Research ArticleThe Relationship between the StochasticMaximum Principle and the Dynamic Programming inSingular Control of Jump Diffusions

Farid Chighoub and Brahim Mezerdi

Laboratory of Applied Mathematics University Mohamed Khider PO Box 145 07000 Biskra Algeria

Correspondence should be addressed to Brahim Mezerdi bmezerdiyahoofr

Received 7 September 2013 Revised 28 November 2013 Accepted 3 December 2013 Published 9 January 2014

Academic Editor Agnes Sulem

Copyright copy 2014 F Chighoub and B Mezerdi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The main objective of this paper is to explore the relationship between the stochastic maximum principle (SMP in short) anddynamic programming principle (DPP in short) for singular control problems of jump diffusions First we establish necessaryas well as sufficient conditions for optimality by using the stochastic calculus of jump diffusions and some properties of singularcontrols Then we give under smoothness conditions a useful verification theorem and we show that the solution of the adjointequation coincides with the spatial gradient of the value function evaluated along the optimal trajectory of the state equationFinally using these theoretical results we solve explicitly an example on optimal harvesting strategy for a geometric Brownianmotion with jumps

1 Introduction

In this paper we consider a mixed classical-singular controlproblem in which the state evolves according to a stochasticdifferential equation driven by a Poisson random measureand an independent multidimensional Brownian motion ofthe following form

119889119909119905= 119887 (119905 119909

119905 119906

119905) 119889119905 + 120590 (119905 119909

119905 119906

119905) 119889119861

119905

+ int119864

120574 (119905 119909119905minus 119906

119905 119890) (119889119905 119889119890) + 119866

119905119889120585

119905

1199090= 119909

(1)

where 119887 120590 120574 and 119866 are given deterministic functions and 119909is the initial state The control variable is a suitable process(119906 120585) where 119906 [0 119879] times Ω rarr 119860

1sub R119889 is the usual

classical absolutely continuous control and 120585 [0 119879] times Ω rarr

1198602= ([0infin))

119898 is the singular control which is an increasing

process continuous on the right with limits on the left with1205850minus= 0 The performance functional has the form

119869 (119906 120585) = 119864 [int

119879

0

119891 (119905 119909119905 119906

119905) 119889119905 + int

119879

0

119896 (119905) 119889120585119905+ 119892 (119909

119879)]

(2)

The objective of the controller is to choose a couple(119906

⋆ 120585

⋆) of adapted processes in order to maximize the

performance functionalIn the first part of our present work we investigate

the question of necessary as well as sufficient optimalityconditions in the form of a Pontryagin stochastic maximumprinciple In the second part we give under regularityassumptions a useful verification theorem Then we showthat the adjoint process coincides with the spatial gradient ofthe value function evaluated along the optimal trajectory ofthe state equation Finally using these theoretical results wesolve explicitly an example on optimal harvesting strategyfor a geometric Brownian motion with jumps Note thatour results improve those in [1 2] to the jump diffusionsetting Moreover we generalize results in [3 4] by allowing

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2014 Article ID 201491 17 pageshttpdxdoiorg1011552014201491

2 International Journal of Stochastic Analysis

both classical and singular controls at least in the completeinformation setting Note that in our control problem thereare two types of jumps for the state process the inaccessibleones which come from the Poisson martingale part andthe predictable ones which come from the singular controlpart The inclusion of these jump terms introduces a majordifference with respect to the case without singular control

Stochastic control problems of singular type have receivedconsiderable attention due to their wide applicability ina number of different areas see [4ndash8] In most casesthe optimal singular control problem was studied throughdynamic programming principle see [9] where it was shownin particular that the value function is continuous and is theunique viscosity solution of the HJB variational inequality

The one-dimensional problems of the singular typewithout the classical control have been studied by manyauthors It was shown that the value function satisfies avariational inequality which gives rise to a free boundaryproblem and the optimal state process is a diffusion reflectedat the free boundary Bather and Chernoff [10] were the firstto formulate such a problem Benes et al [11] explicitly solveda one-dimensional example by observing that the valuefunction in their example is twice continuously differentiableThis regularity property is called the principle of smooth fitThe optimal control can be constructed by using the reflectedBrownian motion see Lions and Sznitman [12] for moredetails Applications to irreversible investment industryequilibrium and portfolio optimization under transactioncosts can be found in [13] A problem of optimal harvestingfrom a population in a stochastic crowded environment isproposed in [14] to represent the size of the population attime 119905 as the solution of the stochastic logistic differentialequation The two-dimensional problem that arises in port-folio selection models under proportional transaction costsis of singular type and has been considered by Davis andNorman [15] The case of diffusions with jumps is studiedby Oslashksendal and Sulem [8] For further contributions onsingular control problems and their relationshipwith optimalstopping problems the reader is referred to [4 5 7 16 17]

The stochastic maximum principle is another power-ful tool for solving stochastic control problems The firstresult that covers singular control problems was obtainedby Cadenillas and Haussmann [18] in which they considerlinear dynamics convex cost criterion and convex stateconstraints A first-orderweak stochasticmaximumprinciplewas developed via convex perturbations method for bothabsolutely continuous and singular components by Bahlaliand Chala [1] The second-order stochastic maximum prin-ciple for nonlinear SDEs with a controlled diffusion matrixwas obtained by Bahlali and Mezerdi [19] extending thePeng maximum principle [20] to singular control problemsA similar approach has been used by Bahlali et al in [21] tostudy the stochastic maximum principle in relaxed-singularoptimal control in the case of uncontrolled diffusion Bahlaliet al in [22] discuss the stochastic maximum principle insingular optimal control in the case where the coefficientsare Lipschitz continuous in 119909 provided that the classicalderivatives are replaced by the generalized ones See also therecent paper by Oslashksendal and Sulem [4] where Malliavin

calculus techniques have been used to define the adjointprocess

Stochastic control problems in which the system isgoverned by a stochastic differential equation with jumpswithout the singular part have been also studied both bythe dynamic programming approach and by the Pontryaginmaximum principle The HJB equation associated with thisproblems is a nonlinear second-order parabolic integro-differential equation Pham [23] studied a mixed optimalstopping and stochastic control of jump diffusion processesby using the viscosity solutions approach Some verificationtheorems of various types of problems for systems governedby this kind of SDEs are discussed by Oslashksendal and Sulem[8] Some results that cover the stochasticmaximumprinciplefor controlled jump diffusion processes are discussed in [324 25] In [3] the sufficient maximum principle and thelink with the dynamic programming principle are givenby assuming the smoothness of the value function Let usmention that in [24] the verification theorem is establishedin the framework of viscosity solutions and the relation-ship between the adjoint processes and some generalizedgradients of the value function are obtained Note that Shiand Wu [24] extend the results of [26] to jump diffusionsSee also [27] for systematic study of the continuous caseThe second-order stochastic maximum principle for optimalcontrols of nonlinear dynamics with jumps and convex stateconstraints was developed via spike variation method byTang and Li [25] These conditions are described in terms oftwo adjoint processes which are linear backward SDEs Suchequations have important applications in hedging problems[28] Existence and uniqueness for solutions to BSDEs withjumps and nonlinear coefficients have been treated by Tangand Li [25] and Barles et al [29]The linkwith integral-partialdifferential equations is studied in [29]

The plan of the paper is as follows In Section 2 wegive some preliminary results and notations The purpose ofSection 3 is to derive necessary as well as sufficient optimalityconditions In Section 4 we give under-regularity assump-tions a verification theorem for the value function Then weprove that the adjoint process is equal to the derivative of thevalue function evaluated at the optimal trajectory extendingin particular [2 3] An example has been solved explicitly byusing the theoretical results

2 Assumptions and Problem Formulation

The purpose of this section is to introduce some notationswhich will be needed in the subsequent sections In all whatfollows we are given a probability space (ΩF (F

119905)119905le119879P)

such that F0contains the P-null sets F

119879= F for an

arbitrarily fixed time horizon 119879 and (F119905)119905le119879

satisfies theusual conditions We assume that (F

119905)119905le119879

is generated by a119889-dimensional standard Brownianmotion119861 and an indepen-dent jump measure 119873 of a Levy process 120578 on [0 119879] times 119864where 119864 sub R119898

0 for some 119898 ge 1 We denote by (F119861

119905)119905le119879

(resp (F119873

119905)119905le119879

) the P-augmentation of the natural filtrationof 119861 (resp119873) We assume that the compensator of119873 has theform 120583(119889119905 119889119890) = ](119889119890)119889119905 for some 120590-finite Levy measure ]on 119864 endowed with its Borel 120590-fieldB(119864) We suppose that

International Journal of Stochastic Analysis 3

int1198641and |119890|

2](119889119890) lt infin and set (119889119905 119889119890) = 119873(119889119905 119889119890) minus ](119889119890)119889119905for the compensated jumpmartingale randommeasure of119873

Obviously we have

F119905= 120590 [intint

119860times(0119904]

119873(119889119903 119889119890) 119904 le 119905 119860 isinB (119864)]

or 120590 [119861119904 119904 le 119905] orN

(3)

whereN denotes the totality of ]-null sets and1205901or 120590

2denotes

the 120590-field generated by 1205901cup 120590

2

Notation Any element 119909 isin R119899 will be identified with acolumn vector with 119899 components and its norm is |119909| =|119909

1| + sdot sdot sdot + |119909

119899| The scalar product of any two vectors 119909 and

119910 on R119899 is denoted by 119909119910 or sum119899

119894=1119909119894119910119894 For a function ℎ we

denote by ℎ119909(resp ℎ

119909119909) the gradient or Jacobian (resp the

Hessian) of ℎ with respect to the variable 119909Given 119904 lt 119905 let us introduce the following spaces

(i) L2

](119864R119899) or L2

] is the set of square integrable functionsl(sdot) 119864 rarr R119899 such that

l (119890)2L2](119864R119899)

= int119864

|l (119890)|2] (119889119890) lt infin (4)

(ii) S2

([119904119905]R119899) is the set of R119899-valued adapted cadlagprocesses 119875 such that

119875S2([119904119905]R119899)

= E[ sup119903isin[119904119905]

100381610038161003816100381611987511990310038161003816100381610038162

]

12

lt infin (5)

(iii) M2

([119904119905]R119899) is the set of progressively measurable R119899-valued processes 119876 such that

119876M2([119904119905]R119899)

= E[int119905

119904

1003816100381610038161003816119876119903

10038161003816100381610038162

119889119903]

12

lt infin (6)

(iv) L2

]([119904119905]R119899) is the set of B([0 119879] times Ω) otimes B(119864)

measurable maps 119877 [0 119879] times Ω times 119864 rarr R119899 suchthat

119877L2]([119904119905]R119899)

= E[int119905

119904

int119864

1003816100381610038161003816119877119903(119890)10038161003816100381610038162] (119889119890) 119889119903]

12

lt infin (7)

To avoid heavy notations we omit the subscript([119904 119905]R119899

) in these notations when (119904 119905) = (0 119879)Let 119879 be a fixed strictly positive real number 119860

1is a

closed convex subset ofR119899 and1198602= ([0infin)

119898) Let us define

the class of admissible control processes (119906 120585)

Definition 1 An admissible control is a pair of measurableadapted processes 119906 [0 119879]timesΩ rarr 119860

1 and 120585 [0 119879]timesΩ rarr

1198602 such that

(1) 119906 is a predictable process 120585 is of bounded variationnondecreasing right continuous with left-hand lim-its and 120585

0minus= 0

(2) E[sup119905isin[0119879]

|119906119905|2+ |120585

119879|2] lt infin

We denote by U = U1times U

2the set of all admissible

controls Here U1(resp U

2) represents the set of the

admissible controls 119906 (resp 120585)

Assume that for (119906 120585) isin U 119905 isin [0 119879] the state 119909119905of our

system is given by

119889119909119905= 119887 (119905 119909

119905 119906

119905) 119889119905 + 120590 (119905 119909

119905 119906

119905) 119889119861

119905

+ int119864

120574 (119905 119909119905minus 119906

119905 119890) (119889119905 119889119890) + 119866

119905119889120585

119905

1199090= 119909

(8)

where 119909 isin R119899 is given representing the initial stateLet

119887 [0 119879] timesR119899times 119860

1997888rarr R

119899

120590 [0 119879] timesR119899times 119860

1997888rarr R

119899times119889

120574 [0 119879] timesR119899times 119860

1times 119864 997888rarr R

119899

119866 [0 119879] 997888rarr R119899times119898

(9)

be measurable functionsNotice that the jump of a singular control 120585 isin U

2at any

jumping time 120591 is defined by Δ120585120591= 120585

120591minus 120585

120591minus and we let

120585119888

119905= 120585

119905minus sum

0lt120591le119905

Δ120585120591 (10)

be the continuous part of 120585We distinguish between the jumps of 119909

120591caused by the

jump of119873(120591 119890) defined by

Δ119873119909120591= int

119864

120574 (120591 119909120591minus 119906

120591 119890)119873 (120591 119889119890)

= 120574 (120591 119909

120591minus 119906

120591 119890) if 120578 has a jump of size 119890 at 120591

0 otherwise(11)

and the jump of 119909120591caused by the singular control 120585 denoted

by Δ120585119909120591= 119866

120591Δ120585

120591 In the above 119873(120591 sdot) represents the

jump in the Poisson randommeasure occurring at time 120591 Inparticular the general jump of the state process at 120591 is givenby Δ119909

120591= 119909

120591minus 119909

120591minus= Δ

120585119909120591+ Δ

119873119909120591

If 120593 is a continuous real function we let

Δ120585120593 (119909

120591) = 120593 (119909

120591) minus 120593 (119909

120591minus+ Δ

119873119909120591) (12)

The expression (12) defines the jump in the value of120593(119909

120591) caused by the jump of 119909 at 120591 We emphasize that the

possible jumps in 119909120591coming from the Poisson measure are

not included in Δ120585120593(119909

120591)

Suppose that the performance functional has the form

119869 (119906 120585) = E [int119879

0

119891 (119905 119909119905 119906

119905) 119889119905 + 119892 (119909

119879) + int

119879

119904

119896119905119889120585

119905]

for (119906 120585) isin U(13)

4 International Journal of Stochastic Analysis

where 119891 [0 119879] times R119899times 119860

1rarr R 119892 R119899

rarr R and 119896 [0 119879] rarr ([0infin))

119898 with 119896119905119889120585

119905= sum

119898

119897=1119896119897

119905119889120585

119897

119905

An admissible control (119906⋆ 120585⋆) is optimal if

119869 (119906⋆ 120585

⋆) = sup

(119906120585)isinU

119869 (119906 120585) (14)

Let us assume the following

(H1) Themaps 119887120590 120574 and119891 are continuously differentiablewith respect to (119909 119906) and 119892 is continuously differen-tiable in 119909

(H2) The derivatives 119887

119909 119887

119906 120590

119909 120590

119906 120574

119909 120574

119906 119891

119909 119891

119906 and 119892

119909are

continuous in (119909 119906) and uniformly bounded

(H3) 119887 120590 120574 and 119891 are bounded by119870

1(1 + |119909| + |119906|) and 119892

is bounded by 1198701(1 + |119909|) for some119870

1gt 0

(H4) For all (119906 119890) isin 119860

1times 119864 the map

(119909 120577) isin R119899timesR

119899997888rarr 119886 (119905 119909 119906 120577 119890)

= 120577T(120574

119909(119905 119909 119906 119890) + 119868

119889) 120577

(15)

satisfies uniformly in (119909 120577) isin R119899timesR119899

119886 (119905 119909 119906 120577 119890) ge100381610038161003816100381612057710038161003816100381610038162

119870minus1

2 for some 119870

2gt 0 (16)

(H5) 119866 119896 are continuous and bounded

3 The Stochastic Maximum Principle

Let us first define the usual Hamiltonian associated to thecontrol problem by

119867(119905 119909 119906 119901 119902X (sdot)) = 119891 (119905 119909 119906) + 119901119887 (119905 119909 119906)

+

119899

sum

119895=1

119902119895120590119895(119905 119909 119906)

+ int119864

X (119890) 120574 (119905 119909 119906 119890) ] (119889119890)

(17)

where (119905 119909 119906 119901 119902X(sdot)) isin [0 119879]timesR119899times119860

1timesR119899

timesR119899times119899timesL2

] 119902119895

and 120590119895 for 119895 = 1 119899 denote the 119895th column of the matrices119902 and 120590 respectively

Let (119906⋆ 120585⋆) be an optimal control and let 119909⋆ be thecorresponding optimal trajectory Then we consider a triple(119901 119902 119903(sdot)) of square integrable adapted processes associatedwith (119906⋆ 119909⋆) with values in R119899

timesR119899times119889timesR119899 such that

119889119901119905= minus119867

119909(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119905119889119861

119905+ int

119864

119903119905(119890) (119889119905 119889119890)

119901119879= 119892

119909(119909

⋆

119879)

(18)

31 Necessary Conditions of Optimality The purpose of thissection is to derive optimality necessary conditions satisfiedby an optimal control assuming that the solution exists Theproof is based on convex perturbations for both absolutelycontinuous and singular components of the optimal controland on some estimates of the state processes Note that ourresults generalize [1 2 21] for systems with jumps

Theorem 2 (necessary conditions of optimality) Let (119906⋆ 120585⋆)be an optimal control maximizing the functional 119869 overU andlet 119909⋆ be the corresponding optimal trajectoryThen there existsan adapted process (119901 119902 119903(sdot)) isin S2

times M2times L2

] which isthe unique solution of the BSDE (18) such that the followingconditions hold

(i) For all V isin 1198601

119867119906(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) le 0

119889119905mdash119886119890 Pmdash119886119904(19)

(ii) For all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (20)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (21)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (22)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (23)

where Δ119873119901119905= int

119864119903119905(119890)119873(119905 119889119890)

In order to prove Theorem 2 we present some auxiliaryresults

311 Variational Equation Let (V 120585) isin U be such that (119906⋆ +V 120585⋆+120585) isin UThe convexity condition of the control domainensures that for 120576 isin (0 1) the control (119906⋆+120576V 120585⋆+120576120585) is also inUWe denote by119909120576 the solution of the SDE (8) correspondingto the control (119906⋆ + 120576V 120585⋆ + 120576120585) Then by standard argumentsfrom stochastic calculus it is easy to check the followingestimate

Lemma 3 Under assumptions (H1)ndash(H

5) one has

lim120576rarr0

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

] = 0 (24)

Proof From assumptions (H1)ndash(H

5) we get by using the

Burkholder-Davis-Gundy inequality

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

]

le 119870int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816119909120576

120591minus 119909

⋆

120591

10038161003816100381610038162

]119889119904

+1198701205762(int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816V12059110038161003816100381610038162

]119889119904 + E100381610038161003816100381612058511987910038161003816100381610038162

)

(25)

International Journal of Stochastic Analysis 5

From Definition 1 and Gronwallrsquos lemma the result fol-lows immediately by letting 120576 go to zero

We define the process 119911119905= 119911

119906⋆

V120585119905

by

119889119911119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905 119889119905

+

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

t ) V119905 119889119861119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905

times (119889119905 119889119890) + 119866119905119889120585

119905

1199110= 0

(26)

From (H2) and Definition 1 one can find a unique

solution 119911which solves the variational equation (26) and thefollowing estimate holds

Lemma 4 Under assumptions (H1)ndash(H

5) it holds that

lim120576rarr0

E

100381610038161003816100381610038161003816100381610038161003816

119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905

100381610038161003816100381610038161003816100381610038161003816

2

= 0 (27)

Proof Let

Γ120576

119905=119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905 (28)

We denote 119909120583120576119905= 119909

⋆

119905+ 120583120576(Γ

120576

119905+ 119911

119905) and 119906120583120576

119905= 119906

⋆

119905+ 120583120576V

119905

for notational convenience Then we have immediately thatΓ120576

0= 0 and Γ120576

119905satisfies the following SDE

119889Γ120576

119905= 1

120576(119887 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 119887 (119905 119909

⋆

119905 119906

⋆

119905))

minus (119887119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119905

+ 1

120576(120590 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 120590 (119905 119909

⋆

119905 119906

⋆

119905))

minus (120590119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119861

119905

+ int119864

1

120576(120574 (119905 119909

120583120576

119905minus 119906

120583120576

119905 119890) minus 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890))

minus (120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905)

times (119889119905 119889119890)

(29)

Since the derivatives of the coefficients are bounded andfrom Definition 1 it is easy to verify by Gronwallrsquos inequalitythat Γ120576 isin S2 and

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

int119864

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120574119909(119904 119909

120583120576

119904 119906

120583120576

119904 119890) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

] (119889119890) 119889119904

+ 119870E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(30)

where 120588120576119905is given by

120588120576

119905= minus int

119905

0

119887119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119904 minus int

119905

0

120590119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119861

119904

minus int

119905

0

int119864

120574119909(119904 119909

⋆

119904minus 119906

⋆

119904 119890) 119911

119904minus (119889119904 119889119890)

minus int

119905

0

119887V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119904 minus int

119905

0

120590V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119861

119904

minus int

119905

0

int119864

120574V (119904 119909⋆

119904minus 119906

⋆

119904 119890) V

119904 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574119909(119904 119909

120583120576

119904minus 119906

120583120576

119904 119890) 119911

119904minus119889120583 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887V (119904 119909120583120576

119904 119906

120583120576

119904) V

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590V (119904 119909120583120576

s 119906120583120576

119904) V

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574V (119904 119909120583120576

119904minus 119906

120583120576

119904 119890) V

119904119889120583 (119889119904 119889119890)

(31)

Since 119887119909 120590

119909 and 120574

119909are bounded then

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119872Eint119905

0

1003816100381610038161003816Γ120576

119904

10038161003816100381610038162

119889119904 +119872E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(32)

where119872 is a generic constant depending on the constants119870](119864) and 119879 We conclude from Lemma 3 and the dominatedconvergence theorem that lim

120576rarr0120588120576

119905= 0 Hence (27)

follows from Gronwallrsquos lemma and by letting 120576 go to 0 Thiscompletes the proof

312 Variational Inequality Let Φ be the solution of thelinear matrix equation for 0 le 119904 lt 119905 le 119879

119889Φ119904119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119905 +

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119861

119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) Φ

119904119905minus (119889119905 119889119890)

Φ119904119904= 119868

119889

(33)

where 119868119889is the 119899 times 119899 identity matrix This equation is linear

with bounded coefficients then it admits a unique strong

6 International Journal of Stochastic Analysis

solution Moreover the condition (H4) ensures that the

tangent process Φ is invertible with an inverse Ψ satisfyingsuitable integrability conditions

From Itorsquos formula we can easily check that 119889(Φ119904119905Ψ119904119905) =

0 and Φ119904119904Ψ119904119904= 119868

119889 where Ψ is the solution of the following

equation

119889Ψ119904119905= minusΨ

119904119905

119887119909(119905 119909

⋆

119905 119906

⋆

119905) minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119909(119905 119909

⋆

119905 119906

⋆

119905)

minusint119864

120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

119889119905

minus

119889

sum

119895=1

Ψ119904119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119889119861

119895

119905

minus Ψ119904119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

times 119873 (119889119905 119889119890)

Ψ119904119904= 119868

119889

(34)

so Ψ = Φminus1 If 119904 = 0 we simply write Φ0119905= Φ

119905and Ψ

0119905= Ψ

119905

By the integration by parts formula ([8 Lemma 36]) we cansee that the solution of (26) is given by 119911

119905= Φ

119905120578119905 where 120578

119905is

the solution of the stochastic differential equation

119889120578119905= Ψ

119905

119887119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905

minusint119864

120574119906(119905 119909

⋆

119905 119906

⋆

119905 119911) V

119905] (119889119890)

119889119905

+

119889

sum

119895=1

Ψ119905120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119861

119895

119905

+ Ψ119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905119873(119889119905 119889119890)

+ Ψ119905119866

119905119889120585

119905minus Ψ

119905int119864

(120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890)119873 (119905 119889119890) 119866

119905Δ120585

119905

1205780= 0

(35)Let us introduce the following convex perturbation of the

optimal control (119906⋆ 120585⋆) defined by(119906

⋆120576 120585

⋆120576) = (119906

⋆+ 120576V 120585⋆ + 120576120585) (36)

for some (V 120585) isin U and 120576 isin (0 1) Since (119906⋆ 120585⋆) is an optimalcontrol then 120576minus1(119869(119906120576 120585120576) minus 119869(119906⋆ 120585⋆)) le 0 Thus a necessarycondition for optimality is that

lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆)) le 0 (37)

The rest of this subsection is devoted to the computationof the above limitWewill see that the expression (37) leads toa precise description of the optimal control (119906⋆ 120585⋆) in termsof the adjoint process First it is easy to prove the followinglemma

Lemma 5 Under assumptions (H1)ndash(H

5) one has

119868 = lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+ 119892119909(119909

⋆

119879) 119911

119879+int

119879

0

119896119905119889120585

119905]

(38)

Proof Weuse the same notations as in the proof of Lemma 4First we have

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904+ 119891

119906(119904 119909

120583120576

119904 119906

120583120576

119904) V

119904 119889120583 119889119904

+ int

1

0

119892119909(119909

120583120576

119879) 119911

119879119889120583 + int

119879

0

119896119905119889120585

119905] + 120573

120576

119905

(39)

where

120573120576

119905= E [int

119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583 119889119904 + int

1

0

119892119909(119909

120583120576

119879) Γ

120576

119879119889120583]

(40)

By using Lemma 4 and since the derivatives 119891119909 119891

119906 and

119892119909are bounded we have lim

120576rarr0120573120576

119905= 0 Then the result

follows by letting 120576 go to 0 in the above equality

Substituting by 119911119905= Φ

119905120578119905in (38) leads to

119868 = E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904120578119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+119892119909(119909

⋆

119879)Φ

119879120578119879+ int

119879

0

119896119905119889120585

119905]

(41)

Consider the right continuous version of the squareintegrable martingale

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879| F

119905] (42)

By the Ito representation theorem [30] there exist twoprocesses 119876 = (1198761

119876119889) where 119876119895

isinM2 for 119895 = 1 119889and 119880(sdot) isinL2

] satisfying

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879]

+

119889

sum

119895=1

int

119905

0

119876119895

119904119889119861

119895

119904+ int

119905

0

int119864

119880119904(119890) (119889119904 119889119890)

(43)

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Stochastic Analysis

both classical and singular controls at least in the completeinformation setting Note that in our control problem thereare two types of jumps for the state process the inaccessibleones which come from the Poisson martingale part andthe predictable ones which come from the singular controlpart The inclusion of these jump terms introduces a majordifference with respect to the case without singular control

Stochastic control problems of singular type have receivedconsiderable attention due to their wide applicability ina number of different areas see [4ndash8] In most casesthe optimal singular control problem was studied throughdynamic programming principle see [9] where it was shownin particular that the value function is continuous and is theunique viscosity solution of the HJB variational inequality

The one-dimensional problems of the singular typewithout the classical control have been studied by manyauthors It was shown that the value function satisfies avariational inequality which gives rise to a free boundaryproblem and the optimal state process is a diffusion reflectedat the free boundary Bather and Chernoff [10] were the firstto formulate such a problem Benes et al [11] explicitly solveda one-dimensional example by observing that the valuefunction in their example is twice continuously differentiableThis regularity property is called the principle of smooth fitThe optimal control can be constructed by using the reflectedBrownian motion see Lions and Sznitman [12] for moredetails Applications to irreversible investment industryequilibrium and portfolio optimization under transactioncosts can be found in [13] A problem of optimal harvestingfrom a population in a stochastic crowded environment isproposed in [14] to represent the size of the population attime 119905 as the solution of the stochastic logistic differentialequation The two-dimensional problem that arises in port-folio selection models under proportional transaction costsis of singular type and has been considered by Davis andNorman [15] The case of diffusions with jumps is studiedby Oslashksendal and Sulem [8] For further contributions onsingular control problems and their relationshipwith optimalstopping problems the reader is referred to [4 5 7 16 17]

The stochastic maximum principle is another power-ful tool for solving stochastic control problems The firstresult that covers singular control problems was obtainedby Cadenillas and Haussmann [18] in which they considerlinear dynamics convex cost criterion and convex stateconstraints A first-orderweak stochasticmaximumprinciplewas developed via convex perturbations method for bothabsolutely continuous and singular components by Bahlaliand Chala [1] The second-order stochastic maximum prin-ciple for nonlinear SDEs with a controlled diffusion matrixwas obtained by Bahlali and Mezerdi [19] extending thePeng maximum principle [20] to singular control problemsA similar approach has been used by Bahlali et al in [21] tostudy the stochastic maximum principle in relaxed-singularoptimal control in the case of uncontrolled diffusion Bahlaliet al in [22] discuss the stochastic maximum principle insingular optimal control in the case where the coefficientsare Lipschitz continuous in 119909 provided that the classicalderivatives are replaced by the generalized ones See also therecent paper by Oslashksendal and Sulem [4] where Malliavin

calculus techniques have been used to define the adjointprocess

Stochastic control problems in which the system isgoverned by a stochastic differential equation with jumpswithout the singular part have been also studied both bythe dynamic programming approach and by the Pontryaginmaximum principle The HJB equation associated with thisproblems is a nonlinear second-order parabolic integro-differential equation Pham [23] studied a mixed optimalstopping and stochastic control of jump diffusion processesby using the viscosity solutions approach Some verificationtheorems of various types of problems for systems governedby this kind of SDEs are discussed by Oslashksendal and Sulem[8] Some results that cover the stochasticmaximumprinciplefor controlled jump diffusion processes are discussed in [324 25] In [3] the sufficient maximum principle and thelink with the dynamic programming principle are givenby assuming the smoothness of the value function Let usmention that in [24] the verification theorem is establishedin the framework of viscosity solutions and the relation-ship between the adjoint processes and some generalizedgradients of the value function are obtained Note that Shiand Wu [24] extend the results of [26] to jump diffusionsSee also [27] for systematic study of the continuous caseThe second-order stochastic maximum principle for optimalcontrols of nonlinear dynamics with jumps and convex stateconstraints was developed via spike variation method byTang and Li [25] These conditions are described in terms oftwo adjoint processes which are linear backward SDEs Suchequations have important applications in hedging problems[28] Existence and uniqueness for solutions to BSDEs withjumps and nonlinear coefficients have been treated by Tangand Li [25] and Barles et al [29]The linkwith integral-partialdifferential equations is studied in [29]

The plan of the paper is as follows In Section 2 wegive some preliminary results and notations The purpose ofSection 3 is to derive necessary as well as sufficient optimalityconditions In Section 4 we give under-regularity assump-tions a verification theorem for the value function Then weprove that the adjoint process is equal to the derivative of thevalue function evaluated at the optimal trajectory extendingin particular [2 3] An example has been solved explicitly byusing the theoretical results

2 Assumptions and Problem Formulation

The purpose of this section is to introduce some notationswhich will be needed in the subsequent sections In all whatfollows we are given a probability space (ΩF (F

119905)119905le119879P)

such that F0contains the P-null sets F

119879= F for an

arbitrarily fixed time horizon 119879 and (F119905)119905le119879

satisfies theusual conditions We assume that (F

119905)119905le119879

is generated by a119889-dimensional standard Brownianmotion119861 and an indepen-dent jump measure 119873 of a Levy process 120578 on [0 119879] times 119864where 119864 sub R119898

0 for some 119898 ge 1 We denote by (F119861

119905)119905le119879

(resp (F119873

119905)119905le119879

) the P-augmentation of the natural filtrationof 119861 (resp119873) We assume that the compensator of119873 has theform 120583(119889119905 119889119890) = ](119889119890)119889119905 for some 120590-finite Levy measure ]on 119864 endowed with its Borel 120590-fieldB(119864) We suppose that

International Journal of Stochastic Analysis 3

int1198641and |119890|

2](119889119890) lt infin and set (119889119905 119889119890) = 119873(119889119905 119889119890) minus ](119889119890)119889119905for the compensated jumpmartingale randommeasure of119873

Obviously we have

F119905= 120590 [intint

119860times(0119904]

119873(119889119903 119889119890) 119904 le 119905 119860 isinB (119864)]

or 120590 [119861119904 119904 le 119905] orN

(3)

whereN denotes the totality of ]-null sets and1205901or 120590

2denotes

the 120590-field generated by 1205901cup 120590

2

Notation Any element 119909 isin R119899 will be identified with acolumn vector with 119899 components and its norm is |119909| =|119909

1| + sdot sdot sdot + |119909

119899| The scalar product of any two vectors 119909 and

119910 on R119899 is denoted by 119909119910 or sum119899

119894=1119909119894119910119894 For a function ℎ we

denote by ℎ119909(resp ℎ

119909119909) the gradient or Jacobian (resp the

Hessian) of ℎ with respect to the variable 119909Given 119904 lt 119905 let us introduce the following spaces

(i) L2

](119864R119899) or L2

] is the set of square integrable functionsl(sdot) 119864 rarr R119899 such that

l (119890)2L2](119864R119899)

= int119864

|l (119890)|2] (119889119890) lt infin (4)

(ii) S2

([119904119905]R119899) is the set of R119899-valued adapted cadlagprocesses 119875 such that

119875S2([119904119905]R119899)

= E[ sup119903isin[119904119905]

100381610038161003816100381611987511990310038161003816100381610038162

]

12

lt infin (5)

(iii) M2

([119904119905]R119899) is the set of progressively measurable R119899-valued processes 119876 such that

119876M2([119904119905]R119899)

= E[int119905

119904

1003816100381610038161003816119876119903

10038161003816100381610038162

119889119903]

12

lt infin (6)

(iv) L2

]([119904119905]R119899) is the set of B([0 119879] times Ω) otimes B(119864)

measurable maps 119877 [0 119879] times Ω times 119864 rarr R119899 suchthat

119877L2]([119904119905]R119899)

= E[int119905

119904

int119864

1003816100381610038161003816119877119903(119890)10038161003816100381610038162] (119889119890) 119889119903]

12

lt infin (7)

To avoid heavy notations we omit the subscript([119904 119905]R119899

) in these notations when (119904 119905) = (0 119879)Let 119879 be a fixed strictly positive real number 119860

1is a

closed convex subset ofR119899 and1198602= ([0infin)

119898) Let us define

the class of admissible control processes (119906 120585)

Definition 1 An admissible control is a pair of measurableadapted processes 119906 [0 119879]timesΩ rarr 119860

1 and 120585 [0 119879]timesΩ rarr

1198602 such that

(1) 119906 is a predictable process 120585 is of bounded variationnondecreasing right continuous with left-hand lim-its and 120585

0minus= 0

(2) E[sup119905isin[0119879]

|119906119905|2+ |120585

119879|2] lt infin

We denote by U = U1times U

2the set of all admissible

controls Here U1(resp U

2) represents the set of the

admissible controls 119906 (resp 120585)

Assume that for (119906 120585) isin U 119905 isin [0 119879] the state 119909119905of our

system is given by

119889119909119905= 119887 (119905 119909

119905 119906

119905) 119889119905 + 120590 (119905 119909

119905 119906

119905) 119889119861

119905

+ int119864

120574 (119905 119909119905minus 119906

119905 119890) (119889119905 119889119890) + 119866

119905119889120585

119905

1199090= 119909

(8)

where 119909 isin R119899 is given representing the initial stateLet

119887 [0 119879] timesR119899times 119860

1997888rarr R

119899

120590 [0 119879] timesR119899times 119860

1997888rarr R

119899times119889

120574 [0 119879] timesR119899times 119860

1times 119864 997888rarr R

119899

119866 [0 119879] 997888rarr R119899times119898

(9)

be measurable functionsNotice that the jump of a singular control 120585 isin U

2at any

jumping time 120591 is defined by Δ120585120591= 120585

120591minus 120585

120591minus and we let

120585119888

119905= 120585

119905minus sum

0lt120591le119905

Δ120585120591 (10)

be the continuous part of 120585We distinguish between the jumps of 119909

120591caused by the

jump of119873(120591 119890) defined by

Δ119873119909120591= int

119864

120574 (120591 119909120591minus 119906

120591 119890)119873 (120591 119889119890)

= 120574 (120591 119909

120591minus 119906

120591 119890) if 120578 has a jump of size 119890 at 120591

0 otherwise(11)

and the jump of 119909120591caused by the singular control 120585 denoted

by Δ120585119909120591= 119866

120591Δ120585

120591 In the above 119873(120591 sdot) represents the

jump in the Poisson randommeasure occurring at time 120591 Inparticular the general jump of the state process at 120591 is givenby Δ119909

120591= 119909

120591minus 119909

120591minus= Δ

120585119909120591+ Δ

119873119909120591

If 120593 is a continuous real function we let

Δ120585120593 (119909

120591) = 120593 (119909

120591) minus 120593 (119909

120591minus+ Δ

119873119909120591) (12)

The expression (12) defines the jump in the value of120593(119909

120591) caused by the jump of 119909 at 120591 We emphasize that the

possible jumps in 119909120591coming from the Poisson measure are

not included in Δ120585120593(119909

120591)

Suppose that the performance functional has the form

119869 (119906 120585) = E [int119879

0

119891 (119905 119909119905 119906

119905) 119889119905 + 119892 (119909

119879) + int

119879

119904

119896119905119889120585

119905]

for (119906 120585) isin U(13)

4 International Journal of Stochastic Analysis

where 119891 [0 119879] times R119899times 119860

1rarr R 119892 R119899

rarr R and 119896 [0 119879] rarr ([0infin))

119898 with 119896119905119889120585

119905= sum

119898

119897=1119896119897

119905119889120585

119897

119905

An admissible control (119906⋆ 120585⋆) is optimal if

119869 (119906⋆ 120585

⋆) = sup

(119906120585)isinU

119869 (119906 120585) (14)

Let us assume the following

(H1) Themaps 119887120590 120574 and119891 are continuously differentiablewith respect to (119909 119906) and 119892 is continuously differen-tiable in 119909

(H2) The derivatives 119887

119909 119887

119906 120590

119909 120590

119906 120574

119909 120574

119906 119891

119909 119891

119906 and 119892

119909are

continuous in (119909 119906) and uniformly bounded

(H3) 119887 120590 120574 and 119891 are bounded by119870

1(1 + |119909| + |119906|) and 119892

is bounded by 1198701(1 + |119909|) for some119870

1gt 0

(H4) For all (119906 119890) isin 119860

1times 119864 the map

(119909 120577) isin R119899timesR

119899997888rarr 119886 (119905 119909 119906 120577 119890)

= 120577T(120574

119909(119905 119909 119906 119890) + 119868

119889) 120577

(15)

satisfies uniformly in (119909 120577) isin R119899timesR119899

119886 (119905 119909 119906 120577 119890) ge100381610038161003816100381612057710038161003816100381610038162

119870minus1

2 for some 119870

2gt 0 (16)

(H5) 119866 119896 are continuous and bounded

3 The Stochastic Maximum Principle

Let us first define the usual Hamiltonian associated to thecontrol problem by

119867(119905 119909 119906 119901 119902X (sdot)) = 119891 (119905 119909 119906) + 119901119887 (119905 119909 119906)

+

119899

sum

119895=1

119902119895120590119895(119905 119909 119906)

+ int119864

X (119890) 120574 (119905 119909 119906 119890) ] (119889119890)

(17)

where (119905 119909 119906 119901 119902X(sdot)) isin [0 119879]timesR119899times119860

1timesR119899

timesR119899times119899timesL2

] 119902119895

and 120590119895 for 119895 = 1 119899 denote the 119895th column of the matrices119902 and 120590 respectively

Let (119906⋆ 120585⋆) be an optimal control and let 119909⋆ be thecorresponding optimal trajectory Then we consider a triple(119901 119902 119903(sdot)) of square integrable adapted processes associatedwith (119906⋆ 119909⋆) with values in R119899

timesR119899times119889timesR119899 such that

119889119901119905= minus119867

119909(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119905119889119861

119905+ int

119864

119903119905(119890) (119889119905 119889119890)

119901119879= 119892

119909(119909

⋆

119879)

(18)

31 Necessary Conditions of Optimality The purpose of thissection is to derive optimality necessary conditions satisfiedby an optimal control assuming that the solution exists Theproof is based on convex perturbations for both absolutelycontinuous and singular components of the optimal controland on some estimates of the state processes Note that ourresults generalize [1 2 21] for systems with jumps

Theorem 2 (necessary conditions of optimality) Let (119906⋆ 120585⋆)be an optimal control maximizing the functional 119869 overU andlet 119909⋆ be the corresponding optimal trajectoryThen there existsan adapted process (119901 119902 119903(sdot)) isin S2

times M2times L2

] which isthe unique solution of the BSDE (18) such that the followingconditions hold

(i) For all V isin 1198601

119867119906(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) le 0

119889119905mdash119886119890 Pmdash119886119904(19)

(ii) For all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (20)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (21)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (22)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (23)

where Δ119873119901119905= int

119864119903119905(119890)119873(119905 119889119890)

In order to prove Theorem 2 we present some auxiliaryresults

311 Variational Equation Let (V 120585) isin U be such that (119906⋆ +V 120585⋆+120585) isin UThe convexity condition of the control domainensures that for 120576 isin (0 1) the control (119906⋆+120576V 120585⋆+120576120585) is also inUWe denote by119909120576 the solution of the SDE (8) correspondingto the control (119906⋆ + 120576V 120585⋆ + 120576120585) Then by standard argumentsfrom stochastic calculus it is easy to check the followingestimate

Lemma 3 Under assumptions (H1)ndash(H

5) one has

lim120576rarr0

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

] = 0 (24)

Proof From assumptions (H1)ndash(H

5) we get by using the

Burkholder-Davis-Gundy inequality

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

]

le 119870int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816119909120576

120591minus 119909

⋆

120591

10038161003816100381610038162

]119889119904

+1198701205762(int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816V12059110038161003816100381610038162

]119889119904 + E100381610038161003816100381612058511987910038161003816100381610038162

)

(25)

International Journal of Stochastic Analysis 5

From Definition 1 and Gronwallrsquos lemma the result fol-lows immediately by letting 120576 go to zero

We define the process 119911119905= 119911

119906⋆

V120585119905

by

119889119911119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905 119889119905

+

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

t ) V119905 119889119861119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905

times (119889119905 119889119890) + 119866119905119889120585

119905

1199110= 0

(26)

From (H2) and Definition 1 one can find a unique

solution 119911which solves the variational equation (26) and thefollowing estimate holds

Lemma 4 Under assumptions (H1)ndash(H

5) it holds that

lim120576rarr0

E

100381610038161003816100381610038161003816100381610038161003816

119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905

100381610038161003816100381610038161003816100381610038161003816

2

= 0 (27)

Proof Let

Γ120576

119905=119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905 (28)

We denote 119909120583120576119905= 119909

⋆

119905+ 120583120576(Γ

120576

119905+ 119911

119905) and 119906120583120576

119905= 119906

⋆

119905+ 120583120576V

119905

for notational convenience Then we have immediately thatΓ120576

0= 0 and Γ120576

119905satisfies the following SDE

119889Γ120576

119905= 1

120576(119887 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 119887 (119905 119909

⋆

119905 119906

⋆

119905))

minus (119887119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119905

+ 1

120576(120590 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 120590 (119905 119909

⋆

119905 119906

⋆

119905))

minus (120590119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119861

119905

+ int119864

1

120576(120574 (119905 119909

120583120576

119905minus 119906

120583120576

119905 119890) minus 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890))

minus (120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905)

times (119889119905 119889119890)

(29)

Since the derivatives of the coefficients are bounded andfrom Definition 1 it is easy to verify by Gronwallrsquos inequalitythat Γ120576 isin S2 and

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

int119864

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120574119909(119904 119909

120583120576

119904 119906

120583120576

119904 119890) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

] (119889119890) 119889119904

+ 119870E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(30)

where 120588120576119905is given by

120588120576

119905= minus int

119905

0

119887119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119904 minus int

119905

0

120590119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119861

119904

minus int

119905

0

int119864

120574119909(119904 119909

⋆

119904minus 119906

⋆

119904 119890) 119911

119904minus (119889119904 119889119890)

minus int

119905

0

119887V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119904 minus int

119905

0

120590V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119861

119904

minus int

119905

0

int119864

120574V (119904 119909⋆

119904minus 119906

⋆

119904 119890) V

119904 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574119909(119904 119909

120583120576

119904minus 119906

120583120576

119904 119890) 119911

119904minus119889120583 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887V (119904 119909120583120576

119904 119906

120583120576

119904) V

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590V (119904 119909120583120576

s 119906120583120576

119904) V

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574V (119904 119909120583120576

119904minus 119906

120583120576

119904 119890) V

119904119889120583 (119889119904 119889119890)

(31)

Since 119887119909 120590

119909 and 120574

119909are bounded then

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119872Eint119905

0

1003816100381610038161003816Γ120576

119904

10038161003816100381610038162

119889119904 +119872E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(32)

where119872 is a generic constant depending on the constants119870](119864) and 119879 We conclude from Lemma 3 and the dominatedconvergence theorem that lim

120576rarr0120588120576

119905= 0 Hence (27)

follows from Gronwallrsquos lemma and by letting 120576 go to 0 Thiscompletes the proof

312 Variational Inequality Let Φ be the solution of thelinear matrix equation for 0 le 119904 lt 119905 le 119879

119889Φ119904119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119905 +

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119861

119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) Φ

119904119905minus (119889119905 119889119890)

Φ119904119904= 119868

119889

(33)

where 119868119889is the 119899 times 119899 identity matrix This equation is linear

with bounded coefficients then it admits a unique strong

6 International Journal of Stochastic Analysis

solution Moreover the condition (H4) ensures that the

tangent process Φ is invertible with an inverse Ψ satisfyingsuitable integrability conditions

From Itorsquos formula we can easily check that 119889(Φ119904119905Ψ119904119905) =

0 and Φ119904119904Ψ119904119904= 119868

119889 where Ψ is the solution of the following

equation

119889Ψ119904119905= minusΨ

119904119905

119887119909(119905 119909

⋆

119905 119906

⋆

119905) minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119909(119905 119909

⋆

119905 119906

⋆

119905)

minusint119864

120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

119889119905

minus

119889

sum

119895=1

Ψ119904119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119889119861

119895

119905

minus Ψ119904119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

times 119873 (119889119905 119889119890)

Ψ119904119904= 119868

119889

(34)

so Ψ = Φminus1 If 119904 = 0 we simply write Φ0119905= Φ

119905and Ψ

0119905= Ψ

119905

By the integration by parts formula ([8 Lemma 36]) we cansee that the solution of (26) is given by 119911

119905= Φ

119905120578119905 where 120578

119905is

the solution of the stochastic differential equation

119889120578119905= Ψ

119905

119887119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905

minusint119864

120574119906(119905 119909

⋆

119905 119906

⋆

119905 119911) V

119905] (119889119890)

119889119905

+

119889

sum

119895=1

Ψ119905120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119861

119895

119905

+ Ψ119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905119873(119889119905 119889119890)

+ Ψ119905119866

119905119889120585

119905minus Ψ

119905int119864

(120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890)119873 (119905 119889119890) 119866

119905Δ120585

119905

1205780= 0

(35)Let us introduce the following convex perturbation of the

optimal control (119906⋆ 120585⋆) defined by(119906

⋆120576 120585

⋆120576) = (119906

⋆+ 120576V 120585⋆ + 120576120585) (36)

for some (V 120585) isin U and 120576 isin (0 1) Since (119906⋆ 120585⋆) is an optimalcontrol then 120576minus1(119869(119906120576 120585120576) minus 119869(119906⋆ 120585⋆)) le 0 Thus a necessarycondition for optimality is that

lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆)) le 0 (37)

The rest of this subsection is devoted to the computationof the above limitWewill see that the expression (37) leads toa precise description of the optimal control (119906⋆ 120585⋆) in termsof the adjoint process First it is easy to prove the followinglemma

Lemma 5 Under assumptions (H1)ndash(H

5) one has

119868 = lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+ 119892119909(119909

⋆

119879) 119911

119879+int

119879

0

119896119905119889120585

119905]

(38)

Proof Weuse the same notations as in the proof of Lemma 4First we have

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904+ 119891

119906(119904 119909

120583120576

119904 119906

120583120576

119904) V

119904 119889120583 119889119904

+ int

1

0

119892119909(119909

120583120576

119879) 119911

119879119889120583 + int

119879

0

119896119905119889120585

119905] + 120573

120576

119905

(39)

where

120573120576

119905= E [int

119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583 119889119904 + int

1

0

119892119909(119909

120583120576

119879) Γ

120576

119879119889120583]

(40)

By using Lemma 4 and since the derivatives 119891119909 119891

119906 and

119892119909are bounded we have lim

120576rarr0120573120576

119905= 0 Then the result

follows by letting 120576 go to 0 in the above equality

Substituting by 119911119905= Φ

119905120578119905in (38) leads to

119868 = E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904120578119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+119892119909(119909

⋆

119879)Φ

119879120578119879+ int

119879

0

119896119905119889120585

119905]

(41)

Consider the right continuous version of the squareintegrable martingale

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879| F

119905] (42)

By the Ito representation theorem [30] there exist twoprocesses 119876 = (1198761

119876119889) where 119876119895

isinM2 for 119895 = 1 119889and 119880(sdot) isinL2

] satisfying

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879]

+

119889

sum

119895=1

int

119905

0

119876119895

119904119889119861

119895

119904+ int

119905

0

int119864

119880119904(119890) (119889119904 119889119890)

(43)

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 3

int1198641and |119890|

2](119889119890) lt infin and set (119889119905 119889119890) = 119873(119889119905 119889119890) minus ](119889119890)119889119905for the compensated jumpmartingale randommeasure of119873

Obviously we have

F119905= 120590 [intint

119860times(0119904]

119873(119889119903 119889119890) 119904 le 119905 119860 isinB (119864)]

or 120590 [119861119904 119904 le 119905] orN

(3)

whereN denotes the totality of ]-null sets and1205901or 120590

2denotes

the 120590-field generated by 1205901cup 120590

2

Notation Any element 119909 isin R119899 will be identified with acolumn vector with 119899 components and its norm is |119909| =|119909

1| + sdot sdot sdot + |119909

119899| The scalar product of any two vectors 119909 and

119910 on R119899 is denoted by 119909119910 or sum119899

119894=1119909119894119910119894 For a function ℎ we

denote by ℎ119909(resp ℎ

119909119909) the gradient or Jacobian (resp the

Hessian) of ℎ with respect to the variable 119909Given 119904 lt 119905 let us introduce the following spaces

(i) L2

](119864R119899) or L2

] is the set of square integrable functionsl(sdot) 119864 rarr R119899 such that

l (119890)2L2](119864R119899)

= int119864

|l (119890)|2] (119889119890) lt infin (4)

(ii) S2

([119904119905]R119899) is the set of R119899-valued adapted cadlagprocesses 119875 such that

119875S2([119904119905]R119899)

= E[ sup119903isin[119904119905]

100381610038161003816100381611987511990310038161003816100381610038162

]

12

lt infin (5)

(iii) M2

([119904119905]R119899) is the set of progressively measurable R119899-valued processes 119876 such that

119876M2([119904119905]R119899)

= E[int119905

119904

1003816100381610038161003816119876119903

10038161003816100381610038162

119889119903]

12

lt infin (6)

(iv) L2

]([119904119905]R119899) is the set of B([0 119879] times Ω) otimes B(119864)

measurable maps 119877 [0 119879] times Ω times 119864 rarr R119899 suchthat

119877L2]([119904119905]R119899)

= E[int119905

119904

int119864

1003816100381610038161003816119877119903(119890)10038161003816100381610038162] (119889119890) 119889119903]

12

lt infin (7)

To avoid heavy notations we omit the subscript([119904 119905]R119899

) in these notations when (119904 119905) = (0 119879)Let 119879 be a fixed strictly positive real number 119860

1is a

closed convex subset ofR119899 and1198602= ([0infin)

119898) Let us define

the class of admissible control processes (119906 120585)

Definition 1 An admissible control is a pair of measurableadapted processes 119906 [0 119879]timesΩ rarr 119860

1 and 120585 [0 119879]timesΩ rarr

1198602 such that

(1) 119906 is a predictable process 120585 is of bounded variationnondecreasing right continuous with left-hand lim-its and 120585

0minus= 0

(2) E[sup119905isin[0119879]

|119906119905|2+ |120585

119879|2] lt infin

We denote by U = U1times U

2the set of all admissible

controls Here U1(resp U

2) represents the set of the

admissible controls 119906 (resp 120585)

Assume that for (119906 120585) isin U 119905 isin [0 119879] the state 119909119905of our

system is given by

119889119909119905= 119887 (119905 119909

119905 119906

119905) 119889119905 + 120590 (119905 119909

119905 119906

119905) 119889119861

119905

+ int119864

120574 (119905 119909119905minus 119906

119905 119890) (119889119905 119889119890) + 119866

119905119889120585

119905

1199090= 119909

(8)

where 119909 isin R119899 is given representing the initial stateLet

119887 [0 119879] timesR119899times 119860

1997888rarr R

119899

120590 [0 119879] timesR119899times 119860

1997888rarr R

119899times119889

120574 [0 119879] timesR119899times 119860

1times 119864 997888rarr R

119899

119866 [0 119879] 997888rarr R119899times119898

(9)

be measurable functionsNotice that the jump of a singular control 120585 isin U

2at any

jumping time 120591 is defined by Δ120585120591= 120585

120591minus 120585

120591minus and we let

120585119888

119905= 120585

119905minus sum

0lt120591le119905

Δ120585120591 (10)

be the continuous part of 120585We distinguish between the jumps of 119909

120591caused by the

jump of119873(120591 119890) defined by

Δ119873119909120591= int

119864

120574 (120591 119909120591minus 119906

120591 119890)119873 (120591 119889119890)

= 120574 (120591 119909

120591minus 119906

120591 119890) if 120578 has a jump of size 119890 at 120591

0 otherwise(11)

and the jump of 119909120591caused by the singular control 120585 denoted

by Δ120585119909120591= 119866

120591Δ120585

120591 In the above 119873(120591 sdot) represents the

jump in the Poisson randommeasure occurring at time 120591 Inparticular the general jump of the state process at 120591 is givenby Δ119909

120591= 119909

120591minus 119909

120591minus= Δ

120585119909120591+ Δ

119873119909120591

If 120593 is a continuous real function we let

Δ120585120593 (119909

120591) = 120593 (119909

120591) minus 120593 (119909

120591minus+ Δ

119873119909120591) (12)

The expression (12) defines the jump in the value of120593(119909

120591) caused by the jump of 119909 at 120591 We emphasize that the

possible jumps in 119909120591coming from the Poisson measure are

not included in Δ120585120593(119909

120591)

Suppose that the performance functional has the form

119869 (119906 120585) = E [int119879

0

119891 (119905 119909119905 119906

119905) 119889119905 + 119892 (119909

119879) + int

119879

119904

119896119905119889120585

119905]

for (119906 120585) isin U(13)

4 International Journal of Stochastic Analysis

where 119891 [0 119879] times R119899times 119860

1rarr R 119892 R119899

rarr R and 119896 [0 119879] rarr ([0infin))

119898 with 119896119905119889120585

119905= sum

119898

119897=1119896119897

119905119889120585

119897

119905

An admissible control (119906⋆ 120585⋆) is optimal if

119869 (119906⋆ 120585

⋆) = sup

(119906120585)isinU

119869 (119906 120585) (14)

Let us assume the following

(H1) Themaps 119887120590 120574 and119891 are continuously differentiablewith respect to (119909 119906) and 119892 is continuously differen-tiable in 119909

(H2) The derivatives 119887

119909 119887

119906 120590

119909 120590

119906 120574

119909 120574

119906 119891

119909 119891

119906 and 119892

119909are

continuous in (119909 119906) and uniformly bounded

(H3) 119887 120590 120574 and 119891 are bounded by119870

1(1 + |119909| + |119906|) and 119892

is bounded by 1198701(1 + |119909|) for some119870

1gt 0

(H4) For all (119906 119890) isin 119860

1times 119864 the map

(119909 120577) isin R119899timesR

119899997888rarr 119886 (119905 119909 119906 120577 119890)

= 120577T(120574

119909(119905 119909 119906 119890) + 119868

119889) 120577

(15)

satisfies uniformly in (119909 120577) isin R119899timesR119899

119886 (119905 119909 119906 120577 119890) ge100381610038161003816100381612057710038161003816100381610038162

119870minus1

2 for some 119870

2gt 0 (16)

(H5) 119866 119896 are continuous and bounded

3 The Stochastic Maximum Principle

Let us first define the usual Hamiltonian associated to thecontrol problem by

119867(119905 119909 119906 119901 119902X (sdot)) = 119891 (119905 119909 119906) + 119901119887 (119905 119909 119906)

+

119899

sum

119895=1

119902119895120590119895(119905 119909 119906)

+ int119864

X (119890) 120574 (119905 119909 119906 119890) ] (119889119890)

(17)

where (119905 119909 119906 119901 119902X(sdot)) isin [0 119879]timesR119899times119860

1timesR119899

timesR119899times119899timesL2

] 119902119895

and 120590119895 for 119895 = 1 119899 denote the 119895th column of the matrices119902 and 120590 respectively

Let (119906⋆ 120585⋆) be an optimal control and let 119909⋆ be thecorresponding optimal trajectory Then we consider a triple(119901 119902 119903(sdot)) of square integrable adapted processes associatedwith (119906⋆ 119909⋆) with values in R119899

timesR119899times119889timesR119899 such that

119889119901119905= minus119867

119909(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119905119889119861

119905+ int

119864

119903119905(119890) (119889119905 119889119890)

119901119879= 119892

119909(119909

⋆

119879)

(18)

31 Necessary Conditions of Optimality The purpose of thissection is to derive optimality necessary conditions satisfiedby an optimal control assuming that the solution exists Theproof is based on convex perturbations for both absolutelycontinuous and singular components of the optimal controland on some estimates of the state processes Note that ourresults generalize [1 2 21] for systems with jumps

Theorem 2 (necessary conditions of optimality) Let (119906⋆ 120585⋆)be an optimal control maximizing the functional 119869 overU andlet 119909⋆ be the corresponding optimal trajectoryThen there existsan adapted process (119901 119902 119903(sdot)) isin S2

times M2times L2

] which isthe unique solution of the BSDE (18) such that the followingconditions hold

(i) For all V isin 1198601

119867119906(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) le 0

119889119905mdash119886119890 Pmdash119886119904(19)

(ii) For all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (20)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (21)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (22)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (23)

where Δ119873119901119905= int

119864119903119905(119890)119873(119905 119889119890)

In order to prove Theorem 2 we present some auxiliaryresults

311 Variational Equation Let (V 120585) isin U be such that (119906⋆ +V 120585⋆+120585) isin UThe convexity condition of the control domainensures that for 120576 isin (0 1) the control (119906⋆+120576V 120585⋆+120576120585) is also inUWe denote by119909120576 the solution of the SDE (8) correspondingto the control (119906⋆ + 120576V 120585⋆ + 120576120585) Then by standard argumentsfrom stochastic calculus it is easy to check the followingestimate

Lemma 3 Under assumptions (H1)ndash(H

5) one has

lim120576rarr0

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

] = 0 (24)

Proof From assumptions (H1)ndash(H

5) we get by using the

Burkholder-Davis-Gundy inequality

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

]

le 119870int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816119909120576

120591minus 119909

⋆

120591

10038161003816100381610038162

]119889119904

+1198701205762(int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816V12059110038161003816100381610038162

]119889119904 + E100381610038161003816100381612058511987910038161003816100381610038162

)

(25)

International Journal of Stochastic Analysis 5

From Definition 1 and Gronwallrsquos lemma the result fol-lows immediately by letting 120576 go to zero

We define the process 119911119905= 119911

119906⋆

V120585119905

by

119889119911119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905 119889119905

+

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

t ) V119905 119889119861119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905

times (119889119905 119889119890) + 119866119905119889120585

119905

1199110= 0

(26)

From (H2) and Definition 1 one can find a unique

solution 119911which solves the variational equation (26) and thefollowing estimate holds

Lemma 4 Under assumptions (H1)ndash(H

5) it holds that

lim120576rarr0

E

100381610038161003816100381610038161003816100381610038161003816

119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905

100381610038161003816100381610038161003816100381610038161003816

2

= 0 (27)

Proof Let

Γ120576

119905=119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905 (28)

We denote 119909120583120576119905= 119909

⋆

119905+ 120583120576(Γ

120576

119905+ 119911

119905) and 119906120583120576

119905= 119906

⋆

119905+ 120583120576V

119905

for notational convenience Then we have immediately thatΓ120576

0= 0 and Γ120576

119905satisfies the following SDE

119889Γ120576

119905= 1

120576(119887 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 119887 (119905 119909

⋆

119905 119906

⋆

119905))

minus (119887119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119905

+ 1

120576(120590 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 120590 (119905 119909

⋆

119905 119906

⋆

119905))

minus (120590119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119861

119905

+ int119864

1

120576(120574 (119905 119909

120583120576

119905minus 119906

120583120576

119905 119890) minus 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890))

minus (120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905)

times (119889119905 119889119890)

(29)

Since the derivatives of the coefficients are bounded andfrom Definition 1 it is easy to verify by Gronwallrsquos inequalitythat Γ120576 isin S2 and

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

int119864

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120574119909(119904 119909

120583120576

119904 119906

120583120576

119904 119890) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

] (119889119890) 119889119904

+ 119870E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(30)

where 120588120576119905is given by

120588120576

119905= minus int

119905

0

119887119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119904 minus int

119905

0

120590119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119861

119904

minus int

119905

0

int119864

120574119909(119904 119909

⋆

119904minus 119906

⋆

119904 119890) 119911

119904minus (119889119904 119889119890)

minus int

119905

0

119887V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119904 minus int

119905

0

120590V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119861

119904

minus int

119905

0

int119864

120574V (119904 119909⋆

119904minus 119906

⋆

119904 119890) V

119904 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574119909(119904 119909

120583120576

119904minus 119906

120583120576

119904 119890) 119911

119904minus119889120583 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887V (119904 119909120583120576

119904 119906

120583120576

119904) V

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590V (119904 119909120583120576

s 119906120583120576

119904) V

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574V (119904 119909120583120576

119904minus 119906

120583120576

119904 119890) V

119904119889120583 (119889119904 119889119890)

(31)

Since 119887119909 120590

119909 and 120574

119909are bounded then

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119872Eint119905

0

1003816100381610038161003816Γ120576

119904

10038161003816100381610038162

119889119904 +119872E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(32)

where119872 is a generic constant depending on the constants119870](119864) and 119879 We conclude from Lemma 3 and the dominatedconvergence theorem that lim

120576rarr0120588120576

119905= 0 Hence (27)

follows from Gronwallrsquos lemma and by letting 120576 go to 0 Thiscompletes the proof

312 Variational Inequality Let Φ be the solution of thelinear matrix equation for 0 le 119904 lt 119905 le 119879

119889Φ119904119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119905 +

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119861

119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) Φ

119904119905minus (119889119905 119889119890)

Φ119904119904= 119868

119889

(33)

where 119868119889is the 119899 times 119899 identity matrix This equation is linear

with bounded coefficients then it admits a unique strong

6 International Journal of Stochastic Analysis

solution Moreover the condition (H4) ensures that the

tangent process Φ is invertible with an inverse Ψ satisfyingsuitable integrability conditions

From Itorsquos formula we can easily check that 119889(Φ119904119905Ψ119904119905) =

0 and Φ119904119904Ψ119904119904= 119868

119889 where Ψ is the solution of the following

equation

119889Ψ119904119905= minusΨ

119904119905

119887119909(119905 119909

⋆

119905 119906

⋆

119905) minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119909(119905 119909

⋆

119905 119906

⋆

119905)

minusint119864

120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

119889119905

minus

119889

sum

119895=1

Ψ119904119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119889119861

119895

119905

minus Ψ119904119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

times 119873 (119889119905 119889119890)

Ψ119904119904= 119868

119889

(34)

so Ψ = Φminus1 If 119904 = 0 we simply write Φ0119905= Φ

119905and Ψ

0119905= Ψ

119905

By the integration by parts formula ([8 Lemma 36]) we cansee that the solution of (26) is given by 119911

119905= Φ

119905120578119905 where 120578

119905is

the solution of the stochastic differential equation

119889120578119905= Ψ

119905

119887119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905

minusint119864

120574119906(119905 119909

⋆

119905 119906

⋆

119905 119911) V

119905] (119889119890)

119889119905

+

119889

sum

119895=1

Ψ119905120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119861

119895

119905

+ Ψ119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905119873(119889119905 119889119890)

+ Ψ119905119866

119905119889120585

119905minus Ψ

119905int119864

(120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890)119873 (119905 119889119890) 119866

119905Δ120585

119905

1205780= 0

(35)Let us introduce the following convex perturbation of the

optimal control (119906⋆ 120585⋆) defined by(119906

⋆120576 120585

⋆120576) = (119906

⋆+ 120576V 120585⋆ + 120576120585) (36)

for some (V 120585) isin U and 120576 isin (0 1) Since (119906⋆ 120585⋆) is an optimalcontrol then 120576minus1(119869(119906120576 120585120576) minus 119869(119906⋆ 120585⋆)) le 0 Thus a necessarycondition for optimality is that

lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆)) le 0 (37)

The rest of this subsection is devoted to the computationof the above limitWewill see that the expression (37) leads toa precise description of the optimal control (119906⋆ 120585⋆) in termsof the adjoint process First it is easy to prove the followinglemma

Lemma 5 Under assumptions (H1)ndash(H

5) one has

119868 = lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+ 119892119909(119909

⋆

119879) 119911

119879+int

119879

0

119896119905119889120585

119905]

(38)

Proof Weuse the same notations as in the proof of Lemma 4First we have

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904+ 119891

119906(119904 119909

120583120576

119904 119906

120583120576

119904) V

119904 119889120583 119889119904

+ int

1

0

119892119909(119909

120583120576

119879) 119911

119879119889120583 + int

119879

0

119896119905119889120585

119905] + 120573

120576

119905

(39)

where

120573120576

119905= E [int

119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583 119889119904 + int

1

0

119892119909(119909

120583120576

119879) Γ

120576

119879119889120583]

(40)

By using Lemma 4 and since the derivatives 119891119909 119891

119906 and

119892119909are bounded we have lim

120576rarr0120573120576

119905= 0 Then the result

follows by letting 120576 go to 0 in the above equality

Substituting by 119911119905= Φ

119905120578119905in (38) leads to

119868 = E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904120578119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+119892119909(119909

⋆

119879)Φ

119879120578119879+ int

119879

0

119896119905119889120585

119905]

(41)

Consider the right continuous version of the squareintegrable martingale

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879| F

119905] (42)

By the Ito representation theorem [30] there exist twoprocesses 119876 = (1198761

119876119889) where 119876119895

isinM2 for 119895 = 1 119889and 119880(sdot) isinL2

] satisfying

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879]

+

119889

sum

119895=1

int

119905

0

119876119895

119904119889119861

119895

119904+ int

119905

0

int119864

119880119904(119890) (119889119904 119889119890)

(43)

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Stochastic Analysis

where 119891 [0 119879] times R119899times 119860

1rarr R 119892 R119899

rarr R and 119896 [0 119879] rarr ([0infin))

119898 with 119896119905119889120585

119905= sum

119898

119897=1119896119897

119905119889120585

119897

119905

An admissible control (119906⋆ 120585⋆) is optimal if

119869 (119906⋆ 120585

⋆) = sup

(119906120585)isinU

119869 (119906 120585) (14)

Let us assume the following

(H1) Themaps 119887120590 120574 and119891 are continuously differentiablewith respect to (119909 119906) and 119892 is continuously differen-tiable in 119909

(H2) The derivatives 119887

119909 119887

119906 120590

119909 120590

119906 120574

119909 120574

119906 119891

119909 119891

119906 and 119892

119909are

continuous in (119909 119906) and uniformly bounded

(H3) 119887 120590 120574 and 119891 are bounded by119870

1(1 + |119909| + |119906|) and 119892

is bounded by 1198701(1 + |119909|) for some119870

1gt 0

(H4) For all (119906 119890) isin 119860

1times 119864 the map

(119909 120577) isin R119899timesR

119899997888rarr 119886 (119905 119909 119906 120577 119890)

= 120577T(120574

119909(119905 119909 119906 119890) + 119868

119889) 120577

(15)

satisfies uniformly in (119909 120577) isin R119899timesR119899

119886 (119905 119909 119906 120577 119890) ge100381610038161003816100381612057710038161003816100381610038162

119870minus1

2 for some 119870

2gt 0 (16)

(H5) 119866 119896 are continuous and bounded

3 The Stochastic Maximum Principle

Let us first define the usual Hamiltonian associated to thecontrol problem by

119867(119905 119909 119906 119901 119902X (sdot)) = 119891 (119905 119909 119906) + 119901119887 (119905 119909 119906)

+

119899

sum

119895=1

119902119895120590119895(119905 119909 119906)

+ int119864

X (119890) 120574 (119905 119909 119906 119890) ] (119889119890)

(17)

where (119905 119909 119906 119901 119902X(sdot)) isin [0 119879]timesR119899times119860

1timesR119899

timesR119899times119899timesL2

] 119902119895

and 120590119895 for 119895 = 1 119899 denote the 119895th column of the matrices119902 and 120590 respectively

Let (119906⋆ 120585⋆) be an optimal control and let 119909⋆ be thecorresponding optimal trajectory Then we consider a triple(119901 119902 119903(sdot)) of square integrable adapted processes associatedwith (119906⋆ 119909⋆) with values in R119899

timesR119899times119889timesR119899 such that

119889119901119905= minus119867

119909(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119905119889119861

119905+ int

119864

119903119905(119890) (119889119905 119889119890)

119901119879= 119892

119909(119909

⋆

119879)

(18)

31 Necessary Conditions of Optimality The purpose of thissection is to derive optimality necessary conditions satisfiedby an optimal control assuming that the solution exists Theproof is based on convex perturbations for both absolutelycontinuous and singular components of the optimal controland on some estimates of the state processes Note that ourresults generalize [1 2 21] for systems with jumps

Theorem 2 (necessary conditions of optimality) Let (119906⋆ 120585⋆)be an optimal control maximizing the functional 119869 overU andlet 119909⋆ be the corresponding optimal trajectoryThen there existsan adapted process (119901 119902 119903(sdot)) isin S2

times M2times L2

] which isthe unique solution of the BSDE (18) such that the followingconditions hold

(i) For all V isin 1198601

119867119906(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) le 0

119889119905mdash119886119890 Pmdash119886119904(19)

(ii) For all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (20)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (21)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (22)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (23)

where Δ119873119901119905= int

119864119903119905(119890)119873(119905 119889119890)

In order to prove Theorem 2 we present some auxiliaryresults

311 Variational Equation Let (V 120585) isin U be such that (119906⋆ +V 120585⋆+120585) isin UThe convexity condition of the control domainensures that for 120576 isin (0 1) the control (119906⋆+120576V 120585⋆+120576120585) is also inUWe denote by119909120576 the solution of the SDE (8) correspondingto the control (119906⋆ + 120576V 120585⋆ + 120576120585) Then by standard argumentsfrom stochastic calculus it is easy to check the followingestimate

Lemma 3 Under assumptions (H1)ndash(H

5) one has

lim120576rarr0

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

] = 0 (24)

Proof From assumptions (H1)ndash(H

5) we get by using the

Burkholder-Davis-Gundy inequality

E[ sup119905isin[0119879]

1003816100381610038161003816119909120576

119905minus 119909

⋆

119905

10038161003816100381610038162

]

le 119870int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816119909120576

120591minus 119909

⋆

120591

10038161003816100381610038162

]119889119904

+1198701205762(int

119879

0

E[ sup120591isin[0119904]

1003816100381610038161003816V12059110038161003816100381610038162

]119889119904 + E100381610038161003816100381612058511987910038161003816100381610038162

)

(25)

International Journal of Stochastic Analysis 5

From Definition 1 and Gronwallrsquos lemma the result fol-lows immediately by letting 120576 go to zero

We define the process 119911119905= 119911

119906⋆

V120585119905

by

119889119911119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905 119889119905

+

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

t ) V119905 119889119861119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905

times (119889119905 119889119890) + 119866119905119889120585

119905

1199110= 0

(26)

From (H2) and Definition 1 one can find a unique

solution 119911which solves the variational equation (26) and thefollowing estimate holds

Lemma 4 Under assumptions (H1)ndash(H

5) it holds that

lim120576rarr0

E

100381610038161003816100381610038161003816100381610038161003816

119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905

100381610038161003816100381610038161003816100381610038161003816

2

= 0 (27)

Proof Let

Γ120576

119905=119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905 (28)

We denote 119909120583120576119905= 119909

⋆

119905+ 120583120576(Γ

120576

119905+ 119911

119905) and 119906120583120576

119905= 119906

⋆

119905+ 120583120576V

119905

for notational convenience Then we have immediately thatΓ120576

0= 0 and Γ120576

119905satisfies the following SDE

119889Γ120576

119905= 1

120576(119887 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 119887 (119905 119909

⋆

119905 119906

⋆

119905))

minus (119887119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119905

+ 1

120576(120590 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 120590 (119905 119909

⋆

119905 119906

⋆

119905))

minus (120590119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119861

119905

+ int119864

1

120576(120574 (119905 119909

120583120576

119905minus 119906

120583120576

119905 119890) minus 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890))

minus (120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905)

times (119889119905 119889119890)

(29)

Since the derivatives of the coefficients are bounded andfrom Definition 1 it is easy to verify by Gronwallrsquos inequalitythat Γ120576 isin S2 and

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

int119864

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120574119909(119904 119909

120583120576

119904 119906

120583120576

119904 119890) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

] (119889119890) 119889119904

+ 119870E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(30)

where 120588120576119905is given by

120588120576

119905= minus int

119905

0

119887119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119904 minus int

119905

0

120590119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119861

119904

minus int

119905

0

int119864

120574119909(119904 119909

⋆

119904minus 119906

⋆

119904 119890) 119911

119904minus (119889119904 119889119890)

minus int

119905

0

119887V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119904 minus int

119905

0

120590V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119861

119904

minus int

119905

0

int119864

120574V (119904 119909⋆

119904minus 119906

⋆

119904 119890) V

119904 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574119909(119904 119909

120583120576

119904minus 119906

120583120576

119904 119890) 119911

119904minus119889120583 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887V (119904 119909120583120576

119904 119906

120583120576

119904) V

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590V (119904 119909120583120576

s 119906120583120576

119904) V

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574V (119904 119909120583120576

119904minus 119906

120583120576

119904 119890) V

119904119889120583 (119889119904 119889119890)

(31)

Since 119887119909 120590

119909 and 120574

119909are bounded then

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119872Eint119905

0

1003816100381610038161003816Γ120576

119904

10038161003816100381610038162

119889119904 +119872E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(32)

where119872 is a generic constant depending on the constants119870](119864) and 119879 We conclude from Lemma 3 and the dominatedconvergence theorem that lim

120576rarr0120588120576

119905= 0 Hence (27)

follows from Gronwallrsquos lemma and by letting 120576 go to 0 Thiscompletes the proof

312 Variational Inequality Let Φ be the solution of thelinear matrix equation for 0 le 119904 lt 119905 le 119879

119889Φ119904119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119905 +

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119861

119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) Φ

119904119905minus (119889119905 119889119890)

Φ119904119904= 119868

119889

(33)

where 119868119889is the 119899 times 119899 identity matrix This equation is linear

with bounded coefficients then it admits a unique strong

6 International Journal of Stochastic Analysis

solution Moreover the condition (H4) ensures that the

tangent process Φ is invertible with an inverse Ψ satisfyingsuitable integrability conditions

From Itorsquos formula we can easily check that 119889(Φ119904119905Ψ119904119905) =

0 and Φ119904119904Ψ119904119904= 119868

119889 where Ψ is the solution of the following

equation

119889Ψ119904119905= minusΨ

119904119905

119887119909(119905 119909

⋆

119905 119906

⋆

119905) minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119909(119905 119909

⋆

119905 119906

⋆

119905)

minusint119864

120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

119889119905

minus

119889

sum

119895=1

Ψ119904119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119889119861

119895

119905

minus Ψ119904119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

times 119873 (119889119905 119889119890)

Ψ119904119904= 119868

119889

(34)

so Ψ = Φminus1 If 119904 = 0 we simply write Φ0119905= Φ

119905and Ψ

0119905= Ψ

119905

By the integration by parts formula ([8 Lemma 36]) we cansee that the solution of (26) is given by 119911

119905= Φ

119905120578119905 where 120578

119905is

the solution of the stochastic differential equation

119889120578119905= Ψ

119905

119887119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905

minusint119864

120574119906(119905 119909

⋆

119905 119906

⋆

119905 119911) V

119905] (119889119890)

119889119905

+

119889

sum

119895=1

Ψ119905120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119861

119895

119905

+ Ψ119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905119873(119889119905 119889119890)

+ Ψ119905119866

119905119889120585

119905minus Ψ

119905int119864

(120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890)119873 (119905 119889119890) 119866

119905Δ120585

119905

1205780= 0

(35)Let us introduce the following convex perturbation of the

optimal control (119906⋆ 120585⋆) defined by(119906

⋆120576 120585

⋆120576) = (119906

⋆+ 120576V 120585⋆ + 120576120585) (36)

for some (V 120585) isin U and 120576 isin (0 1) Since (119906⋆ 120585⋆) is an optimalcontrol then 120576minus1(119869(119906120576 120585120576) minus 119869(119906⋆ 120585⋆)) le 0 Thus a necessarycondition for optimality is that

lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆)) le 0 (37)

The rest of this subsection is devoted to the computationof the above limitWewill see that the expression (37) leads toa precise description of the optimal control (119906⋆ 120585⋆) in termsof the adjoint process First it is easy to prove the followinglemma

Lemma 5 Under assumptions (H1)ndash(H

5) one has

119868 = lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+ 119892119909(119909

⋆

119879) 119911

119879+int

119879

0

119896119905119889120585

119905]

(38)

Proof Weuse the same notations as in the proof of Lemma 4First we have

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904+ 119891

119906(119904 119909

120583120576

119904 119906

120583120576

119904) V

119904 119889120583 119889119904

+ int

1

0

119892119909(119909

120583120576

119879) 119911

119879119889120583 + int

119879

0

119896119905119889120585

119905] + 120573

120576

119905

(39)

where

120573120576

119905= E [int

119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583 119889119904 + int

1

0

119892119909(119909

120583120576

119879) Γ

120576

119879119889120583]

(40)

By using Lemma 4 and since the derivatives 119891119909 119891

119906 and

119892119909are bounded we have lim

120576rarr0120573120576

119905= 0 Then the result

follows by letting 120576 go to 0 in the above equality

Substituting by 119911119905= Φ

119905120578119905in (38) leads to

119868 = E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904120578119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+119892119909(119909

⋆

119879)Φ

119879120578119879+ int

119879

0

119896119905119889120585

119905]

(41)

Consider the right continuous version of the squareintegrable martingale

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879| F

119905] (42)

By the Ito representation theorem [30] there exist twoprocesses 119876 = (1198761

119876119889) where 119876119895

isinM2 for 119895 = 1 119889and 119880(sdot) isinL2

] satisfying

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879]

+

119889

sum

119895=1

int

119905

0

119876119895

119904119889119861

119895

119904+ int

119905

0

int119864

119880119904(119890) (119889119904 119889119890)

(43)

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 5

From Definition 1 and Gronwallrsquos lemma the result fol-lows immediately by letting 120576 go to zero

We define the process 119911119905= 119911

119906⋆

V120585119905

by

119889119911119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905 119889119905

+

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

t ) V119905 119889119861119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905

times (119889119905 119889119890) + 119866119905119889120585

119905

1199110= 0

(26)

From (H2) and Definition 1 one can find a unique

solution 119911which solves the variational equation (26) and thefollowing estimate holds

Lemma 4 Under assumptions (H1)ndash(H

5) it holds that

lim120576rarr0

E

100381610038161003816100381610038161003816100381610038161003816

119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905

100381610038161003816100381610038161003816100381610038161003816

2

= 0 (27)

Proof Let

Γ120576

119905=119909120576

119905minus 119909

⋆

119905

120576minus 119911

119905 (28)

We denote 119909120583120576119905= 119909

⋆

119905+ 120583120576(Γ

120576

119905+ 119911

119905) and 119906120583120576

119905= 119906

⋆

119905+ 120583120576V

119905

for notational convenience Then we have immediately thatΓ120576

0= 0 and Γ120576

119905satisfies the following SDE

119889Γ120576

119905= 1

120576(119887 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 119887 (119905 119909

⋆

119905 119906

⋆

119905))

minus (119887119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 119887

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119905

+ 1

120576(120590 (119905 119909

120583120576

119905 119906

120583120576

119905) minus 120590 (119905 119909

⋆

119905 119906

⋆

119905))

minus (120590119909(119905 119909

⋆

119905 119906

⋆

119905) 119911

119905+ 120590

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905) 119889119861

119905

+ int119864

1

120576(120574 (119905 119909

120583120576

119905minus 119906

120583120576

119905 119890) minus 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890))

minus (120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) 119911

119905minus+ 120574

119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905)

times (119889119905 119889119890)

(29)

Since the derivatives of the coefficients are bounded andfrom Definition 1 it is easy to verify by Gronwallrsquos inequalitythat Γ120576 isin S2 and

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

119889119904

+ 119870Eint119905

0

int119864

100381610038161003816100381610038161003816100381610038161003816

int

1

0

120574119909(119904 119909

120583120576

119904 119906

120583120576

119904 119890) Γ

120576

119904119889120583

100381610038161003816100381610038161003816100381610038161003816

2

] (119889119890) 119889119904

+ 119870E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(30)

where 120588120576119905is given by

120588120576

119905= minus int

119905

0

119887119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119904 minus int

119905

0

120590119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904119889119861

119904

minus int

119905

0

int119864

120574119909(119904 119909

⋆

119904minus 119906

⋆

119904 119890) 119911

119904minus (119889119904 119889119890)

minus int

119905

0

119887V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119904 minus int

119905

0

120590V (119904 119909⋆

119904 119906

⋆

119904) V

119904119889119861

119904

minus int

119905

0

int119864

120574V (119904 119909⋆

119904minus 119906

⋆

119904 119890) V

119904 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574119909(119904 119909

120583120576

119904minus 119906

120583120576

119904 119890) 119911

119904minus119889120583 (119889119904 119889119890)

+ int

119905

0

int

1

0

119887V (119904 119909120583120576

119904 119906

120583120576

119904) V

119904119889120583 119889119904

+ int

119905

0

int

1

0

120590V (119904 119909120583120576

s 119906120583120576

119904) V

119904119889120583 119889119861

119904

+ int

119905

0

int119864

int

1

0

120574V (119904 119909120583120576

119904minus 119906

120583120576

119904 119890) V

119904119889120583 (119889119904 119889119890)

(31)

Since 119887119909 120590

119909 and 120574

119909are bounded then

E1003816100381610038161003816Γ

120576

119905

10038161003816100381610038162

le 119872Eint119905

0

1003816100381610038161003816Γ120576

119904

10038161003816100381610038162

119889119904 +119872E1003816100381610038161003816120588

120576

119905

10038161003816100381610038162

(32)

where119872 is a generic constant depending on the constants119870](119864) and 119879 We conclude from Lemma 3 and the dominatedconvergence theorem that lim

120576rarr0120588120576

119905= 0 Hence (27)

follows from Gronwallrsquos lemma and by letting 120576 go to 0 Thiscompletes the proof

312 Variational Inequality Let Φ be the solution of thelinear matrix equation for 0 le 119904 lt 119905 le 119879

119889Φ119904119905= 119887

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119905 +

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119904119905119889119861

119895

119905

+ int119864

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) Φ

119904119905minus (119889119905 119889119890)

Φ119904119904= 119868

119889

(33)

where 119868119889is the 119899 times 119899 identity matrix This equation is linear

with bounded coefficients then it admits a unique strong

6 International Journal of Stochastic Analysis

solution Moreover the condition (H4) ensures that the

tangent process Φ is invertible with an inverse Ψ satisfyingsuitable integrability conditions

From Itorsquos formula we can easily check that 119889(Φ119904119905Ψ119904119905) =

0 and Φ119904119904Ψ119904119904= 119868

119889 where Ψ is the solution of the following

equation

119889Ψ119904119905= minusΨ

119904119905

119887119909(119905 119909

⋆

119905 119906

⋆

119905) minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119909(119905 119909

⋆

119905 119906

⋆

119905)

minusint119864

120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

119889119905

minus

119889

sum

119895=1

Ψ119904119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119889119861

119895

119905

minus Ψ119904119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

times 119873 (119889119905 119889119890)

Ψ119904119904= 119868

119889

(34)

so Ψ = Φminus1 If 119904 = 0 we simply write Φ0119905= Φ

119905and Ψ

0119905= Ψ

119905

By the integration by parts formula ([8 Lemma 36]) we cansee that the solution of (26) is given by 119911

119905= Φ

119905120578119905 where 120578

119905is

the solution of the stochastic differential equation

119889120578119905= Ψ

119905

119887119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905

minusint119864

120574119906(119905 119909

⋆

119905 119906

⋆

119905 119911) V

119905] (119889119890)

119889119905

+

119889

sum

119895=1

Ψ119905120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119861

119895

119905

+ Ψ119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905119873(119889119905 119889119890)

+ Ψ119905119866

119905119889120585

119905minus Ψ

119905int119864

(120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890)119873 (119905 119889119890) 119866

119905Δ120585

119905

1205780= 0

(35)Let us introduce the following convex perturbation of the

optimal control (119906⋆ 120585⋆) defined by(119906

⋆120576 120585

⋆120576) = (119906

⋆+ 120576V 120585⋆ + 120576120585) (36)

for some (V 120585) isin U and 120576 isin (0 1) Since (119906⋆ 120585⋆) is an optimalcontrol then 120576minus1(119869(119906120576 120585120576) minus 119869(119906⋆ 120585⋆)) le 0 Thus a necessarycondition for optimality is that

lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆)) le 0 (37)

The rest of this subsection is devoted to the computationof the above limitWewill see that the expression (37) leads toa precise description of the optimal control (119906⋆ 120585⋆) in termsof the adjoint process First it is easy to prove the followinglemma

Lemma 5 Under assumptions (H1)ndash(H

5) one has

119868 = lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+ 119892119909(119909

⋆

119879) 119911

119879+int

119879

0

119896119905119889120585

119905]

(38)

Proof Weuse the same notations as in the proof of Lemma 4First we have

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904+ 119891

119906(119904 119909

120583120576

119904 119906

120583120576

119904) V

119904 119889120583 119889119904

+ int

1

0

119892119909(119909

120583120576

119879) 119911

119879119889120583 + int

119879

0

119896119905119889120585

119905] + 120573

120576

119905

(39)

where

120573120576

119905= E [int

119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583 119889119904 + int

1

0

119892119909(119909

120583120576

119879) Γ

120576

119879119889120583]

(40)

By using Lemma 4 and since the derivatives 119891119909 119891

119906 and

119892119909are bounded we have lim

120576rarr0120573120576

119905= 0 Then the result

follows by letting 120576 go to 0 in the above equality

Substituting by 119911119905= Φ

119905120578119905in (38) leads to

119868 = E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904120578119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+119892119909(119909

⋆

119879)Φ

119879120578119879+ int

119879

0

119896119905119889120585

119905]

(41)

Consider the right continuous version of the squareintegrable martingale

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879| F

119905] (42)

By the Ito representation theorem [30] there exist twoprocesses 119876 = (1198761

119876119889) where 119876119895

isinM2 for 119895 = 1 119889and 119880(sdot) isinL2

] satisfying

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879]

+

119889

sum

119895=1

int

119905

0

119876119895

119904119889119861

119895

119904+ int

119905

0

int119864

119880119904(119890) (119889119904 119889119890)

(43)

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Stochastic Analysis

solution Moreover the condition (H4) ensures that the

tangent process Φ is invertible with an inverse Ψ satisfyingsuitable integrability conditions

From Itorsquos formula we can easily check that 119889(Φ119904119905Ψ119904119905) =

0 and Φ119904119904Ψ119904119904= 119868

119889 where Ψ is the solution of the following

equation

119889Ψ119904119905= minusΨ

119904119905

119887119909(119905 119909

⋆

119905 119906

⋆

119905) minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119909(119905 119909

⋆

119905 119906

⋆

119905)

minusint119864

120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

119889119905

minus

119889

sum

119895=1

Ψ119904119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 119889119861

119895

119905

minus Ψ119904119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

times 119873 (119889119905 119889119890)

Ψ119904119904= 119868

119889

(34)

so Ψ = Φminus1 If 119904 = 0 we simply write Φ0119905= Φ

119905and Ψ

0119905= Ψ

119905

By the integration by parts formula ([8 Lemma 36]) we cansee that the solution of (26) is given by 119911

119905= Φ

119905120578119905 where 120578

119905is

the solution of the stochastic differential equation

119889120578119905= Ψ

119905

119887119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905minus

119889

sum

119895=1

120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) 120590

119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905

minusint119864

120574119906(119905 119909

⋆

119905 119906

⋆

119905 119911) V

119905] (119889119890)

119889119905

+

119889

sum

119895=1

Ψ119905120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119861

119895

119905

+ Ψ119905minusint119864

(120574119909(119905 119909

⋆

119905minus 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119906(119905 119909

⋆

119905minus 119906

⋆

119905 119890) V

119905119873(119889119905 119889119890)

+ Ψ119905119866

119905119889120585

119905minus Ψ

119905int119864

(120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

times 120574119909(119905 119909

⋆

119905 119906

⋆

119905 119890)119873 (119905 119889119890) 119866

119905Δ120585

119905

1205780= 0

(35)Let us introduce the following convex perturbation of the

optimal control (119906⋆ 120585⋆) defined by(119906

⋆120576 120585

⋆120576) = (119906

⋆+ 120576V 120585⋆ + 120576120585) (36)

for some (V 120585) isin U and 120576 isin (0 1) Since (119906⋆ 120585⋆) is an optimalcontrol then 120576minus1(119869(119906120576 120585120576) minus 119869(119906⋆ 120585⋆)) le 0 Thus a necessarycondition for optimality is that

lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆)) le 0 (37)

The rest of this subsection is devoted to the computationof the above limitWewill see that the expression (37) leads toa precise description of the optimal control (119906⋆ 120585⋆) in termsof the adjoint process First it is easy to prove the followinglemma

Lemma 5 Under assumptions (H1)ndash(H

5) one has

119868 = lim120576rarr0

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904) 119911

119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+ 119892119909(119909

⋆

119879) 119911

119879+int

119879

0

119896119905119889120585

119905]

(38)

Proof Weuse the same notations as in the proof of Lemma 4First we have

120576minus1(119869 (119906

120576 120585

120576) minus 119869 (119906

⋆ 120585

⋆))

= E [int119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) 119911

119904+ 119891

119906(119904 119909

120583120576

119904 119906

120583120576

119904) V

119904 119889120583 119889119904

+ int

1

0

119892119909(119909

120583120576

119879) 119911

119879119889120583 + int

119879

0

119896119905119889120585

119905] + 120573

120576

119905

(39)

where

120573120576

119905= E [int

119879

0

int

1

0

119891119909(119904 119909

120583120576

119904 119906

120583120576

119904) Γ

120576

119904119889120583 119889119904 + int

1

0

119892119909(119909

120583120576

119879) Γ

120576

119879119889120583]

(40)

By using Lemma 4 and since the derivatives 119891119909 119891

119906 and

119892119909are bounded we have lim

120576rarr0120573120576

119905= 0 Then the result

follows by letting 120576 go to 0 in the above equality

Substituting by 119911119905= Φ

119905120578119905in (38) leads to

119868 = E [int119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904120578119904+ 119891

119906(119904 119909

⋆

119904 119906

⋆

119904) V

119904 119889119904

+119892119909(119909

⋆

119879)Φ

119879120578119879+ int

119879

0

119896119905119889120585

119905]

(41)

Consider the right continuous version of the squareintegrable martingale

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879| F

119905] (42)

By the Ito representation theorem [30] there exist twoprocesses 119876 = (1198761

119876119889) where 119876119895

isinM2 for 119895 = 1 119889and 119880(sdot) isinL2

] satisfying

119872119905= E [int

119879

0

119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 + 119892

119909(119909

⋆

119879)Φ

119879]

+

119889

sum

119895=1

int

119905

0

119876119895

119904119889119861

119895

119904+ int

119905

0

int119864

119880119904(119890) (119889119904 119889119890)

(43)

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 7

Let us denote 119910⋆119905= 119872

119905minusint

119905

0119891119909(119904 119909

⋆

119904 119906

⋆

119904)Φ

119904119889119904 The adjoint

variable is the process defined by

119901119905= 119910

⋆

119905Ψ119905

119902119895

119905= 119876

119895

119905Ψ119905minus 119901

119905120590119895

119909(119905 119909

⋆

119905 119906

⋆

119905) for 119895 = 1 119889

119903119905(119890) = 119880

119905(119890) Ψ

119905(120574

119909(119905 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

+ 119901119905((120574

119909(119904 119909

⋆

119905 119906

⋆

119905 119890) + 119868

119889)minus1

minus 119868119889)

(44)

Theorem 6 Under assumptions (H1)ndash(H

5) one has

119868 = E[int119879

0

119891119906(119904 119909

⋆

119904 119906

⋆

119904) + 119901

119904119887119906(119904 119909

⋆

119904 119906

⋆

119904)

+

119889

sum

119895=1

119902119895

119904120590119895

119906(119904 119909

⋆

119904 119906

⋆

119904)

+ int119864

119903119904(119911) 120574

119906(119904 119909

⋆

119904 119906

⋆

119904 119890) ] (119889119890) V

119904119889119904

+

119898

sum

119894=1

int

119879

0

119896119894

119904+ 119866

119894

119904119901119904 119889120585

119888119894

119904

+

119898

sum

119894=1

sum

0lt119904le119879

119896119894

119904+ 119866

119894

119904(119901

119904minus+ Δ

119873119901119904) Δ120585

119894

119904]

(45)

Proof From the integration by parts formula ([8 Lemma35]) and by using the definition of 119901

119905 119902

119895

119905for 119895 = 1 119889

and 119903119905(sdot) we can easily check that

119864 [119910119879120578119879]

= E[

[

int

119879

0

119901119905119887119906(119905 119909

⋆

119905 119906

⋆

119905) +

119889

sum

119895=1

119902119895

119904120590119895

119906(119905 119909

⋆

119905 119906

⋆

119905)

+ int119864

119903119905(119890) 120574

119906(119905 119909

⋆

119905 119906

⋆

119905 119890) ] (119889119890)

V119905119889119905

minus int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905) 120578

119905Φ

119905119889119905

+

119898

sum

119894=1

(int

119879

0

119866119894

119905119901119905119889120585

119888119894

119905+ sum

0lt119905le119879

119866119894

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

119894

119905)]

]

(46)

Also we have

119868 = E [119910119879120578119879+ int

119879

0

119891119909(119905 119909

⋆

119905 119906

⋆

119905)Φ

119905120578119905119889119905

+int

119879

0

119891119906(119905 119909

⋆

119905 119906

⋆

119905) V

119905119889119905 + int

119879

0

119896119905119889120585

119905]

(47)

substituting (46) in (47) the result follows

313 Adjoint Equation and Maximum Principle Since (37)is true for all (V 120585) isin U and 119868 le 0 we can easily deduce thefollowing result

Theorem 7 Let (119906⋆ 120585⋆) be the optimal control of the problem(14) and denote by 119909⋆ the corresponding optimal trajectorythen the following inequality holds

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(48)

where the Hamiltonian 119867 is defined by (17) and the adjointvariable (119901 119902119895 119903(sdot)) for 119895 = 1 119889 is given by (44)

Now we are ready to give the proof of Theorem 2

Proof of Theorem 2 (i) Let us assume that (119906⋆ 120585⋆) is anoptimal control for the problem (14) so that inequality (48)is valid for every (V 120585) If we choose 120585 = 120585⋆ in inequality(48) we see that for every measurable F

119905-adapted process

V [0 119879] times Ω rarr 1198601

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] le 0 (49)

For V isin U1define

119860V= (119905 120596) isin [0 119879] times Ω

such that 119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) gt 0

(50)

Obviously 119860V119905isin F

119905 for each 119905 isin [0 119879] Let us define

V isin U1by

V119905(120596) =

V if (119905 120596) isin 119860V119905

119906⋆

119905 otherwise

(51)

If 120582 otimesP(119860V) gt 0 where 120582 denotes the Lebesgue measure

then

E [int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905] gt 0 (52)

which contradicts (49) unless 120582 otimes P(119860V) = 0 Hence the

conclusion follows(ii) If instead we choose V = 119906⋆ in inequality (48) we

obtain that for every measurable F119905-adapted process 120585

[0 119879] times Ω rarr 1198602 the following inequality holds

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(53)

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Stochastic Analysis

In particular for 119894 = 1 119898 we put 120585119894119905= 120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905gt0120582(119905) Since the Lebesgue measure is regular then

the purely discontinuous part (120585119894119905minus 120585

⋆119894

119905)119889

= 0 Obviously therelation (53) can be written as

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889(120585

119894minus 120585

⋆119894)119888

119905

+int

119879

0

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 119889(120585

119894minus 120585

⋆119894)119889

119905]

=

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905gt0119889120582 (119905)] gt 0

(54)

This contradicts (53) unless for every 119894 isin 1 119898 120582 otimesP119896119894

119905+ 119866

119894

119905119901119905gt 0 = 0 This proves (20)

Let us prove (21) Define 119889120585119894119905= 1

119896119894

119905+119866119894

119905119901119905minusgt0119889120585

⋆119894

119905+

1119896119894

119905+119866119894

119905119901119905minusle0119889120585

⋆119889119894

119905 for 119894 = 1 119898 then we have 119889(120585119894 minus 120585⋆119894)119888

119905=

minus1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905 and 119889120585119889119894

119905= 119889120585

⋆119889119894

119905 Hence we can rewrite

(53) as follows

minus

119898

sum

119894=1

E [int119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] gt 0 (55)

By comparing with (53) we get119898

sum

119894=1

E [int119879

0

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905] = 0 (56)

then we conclude that119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 1

119896119894

119905+119866119894

119905119901119905le0119889120585

119888119894

119905= 0 (57)

Expressions (22) and (23) are proved by using the sametechniques First for each 119894 isin 1 119898 and 119905 isin [0 119879]

fixed we define 120585119894119904= 120585

119894

119904+ 120575

119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

where 120575119905

denotes theDirac unitmass at 119905 120575119905is a discretemeasure then

(120585119894

119904minus 120585

119894

119904)119888

= 0 and (120585119894119904minus 120585

119894

119904)119889

= 120575119905(119904)1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0

Hence

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)gt0] gt 0 (58)

which contradicts (53) unless for every 119894 isin 1 119898 and119905 isin [0 119879] we have

P 119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) gt 0 = 0 (59)

Next let 120585 be defined by

119889120585119894

119905= 1

119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)ge0119889120585

⋆119894

119905

+ 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0119889120585

⋆119888119894

119905

(60)

Then the relation (53) can be written as119898

sum

119894=1

E[summinus0lt119905le119879

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] gt 0

(61)

which implies that

E[119898

sum

119894=1

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905)

times 1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905] = 0

(62)

By the fact that 119896119894119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) lt 0 and Δ120585119894

119905ge 0 we get

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)lt0Δ120585

⋆119894

119905= 0 (63)

Thus (23) holds The proof is complete

Now by applying Itorsquos formula to 119910⋆119905Ψ119905 it is easy to check

that the processes defined by relation (44) satisfy BSDE (18)called the adjoint equation

32 Sufficient Conditions of Optimality It is well knownthat in the classical cases (without the singular part of thecontrol) the sufficient condition of optimality is of significantimportance in the stochastic maximum principle in thesense that it allows to compute optimal controls This resultstates that under some concavity conditions maximizing theHamiltonian leads to an optimal control

In this section we focus on proving the sufficient maxi-mumprinciple formixed classical-singular stochastic controlproblems where the state of the system is governed by astochastic differential equation with jumps allowing bothclassical control and singular control

Theorem 8 (sufficient condition of optimality in integralform) Let (119906⋆ 120585⋆) be an admissible control and denote 119909⋆the associated controlled state process Let (119901 119902 119903(sdot)) be theunique solution of 119861119878119863119864 (18) Let one assume that (119909 119906) rarr119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr 119892(119909) are concave functions

Moreover suppose that for all 119905 isin [0 119879] V isin 1198601 and 120585 isin U

2

E[int119879

0

119867V (119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) (V

119905minus 119906

⋆

119905) 119889119905

+ int

119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(64)

Then (119906⋆ 120585⋆) is an optimal control

Proof For convenience we will use the following notationsthroughout the proof

Θ⋆(119905) = Θ (119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot))

Θ (119905) = Θ (119905 119909119905 119906

119905 119901

119905 119902

119905 119903

119905(sdot))

for Θ = 119867119867119909 119867

119906

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 9

120575120601 (119905) = 120601 (119905 119909⋆

119905 119906

⋆

119905) minus 120601 (119905 119909

119905 119906

119905)

for 120601 = 119887 120590 120590119895 119895 = 1 119899 119891

120575120574 (119905 119890) = 120574 (119905 119909⋆

119905 119906

⋆

119905 119890) minus 120574 (119905 119909

119905 119906

119905 119890)

120575120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus 119906

⋆

119905 119890) minus 120574 (119905 119909

119905minus 119906

119905 119890)

(65)

Let (119906 120585) be an arbitrary admissible pair and consider thedifference

119869 (119906⋆ 120585

⋆) minus 119869 (119906 120585)

= E [int119879

0

120575119891 (119905) 119889119905 + int

119879

0

119896119905119889(120585

⋆minus 120585)

119905]

+ E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

(66)

We first note that by concavity of 119892 we conclude that

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [(119909⋆

119879minus 119909

119879) 119892

119909(119909

⋆

119879)] = E [(119909

⋆

119879minus 119909

119879) 119901

119879]

= E [int119879

0

(119909⋆

119905minusminus 119909

119905minus) 119889119901

119905+ int

119879

0

119901119905minus119889 (119909

⋆

119905minus 119909

119905)]

+ E[

[

int

119879

0

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905119889119905

+int

119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890) ]

]

+ E[ sum0lt119905le119879

119866119905(Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(67)

which implies that

119864 [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

(119909⋆

119905minus 119909

119905) (minus119867

⋆

119909(119905)) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

119889119905]

]

+ E [int119879

0

int119864

(120575120574minus(119905 119890)) 119903

119905(119890)119873 (119889119905 119889119890)]

+ E [int119879

0

(119909⋆

119905minus 119909

119905) 119902

119905+ (120575120590 (119905)) 119901

119905 119889119861

119905]

+ E [int119879

0

int119864

(119909⋆

119905minusminus 119909

119905minus) 119903

119905(119890) + 119901

119905minus(120575120574

minus(119905 119890))

times (119889119905 119889119890) ]

+ E[int119879

0

119866119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(68)

By the fact that (119901 119902119895 119903(sdot)) isin S2times M2

times L2

] for 119895 =1 119899 we deduce that the stochastic integrals with respectto the local martingales have zero expectation Due to theconcavity of the Hamiltonian119867 the following holds

E [119892 (119909⋆

119879) minus 119892 (119909

119879)]

ge E [int119879

0

minus (119867⋆(119905) minus 119867 (119905)) + 119867

⋆

119906(119905) (119906

⋆

119905minus 119906

119905) 119889119905]

+ E[

[

int

119879

0

119901119905(120575119887 (119905)) +

119899

sum

119895=1

(120575120590119895(119905)) 119902

119895

119905

+int119864

(120575120574 (119905 119890)) 119903119905(119890) ] (119889119890)

119889119905]

]

+ E[int119879

0

119866119879

119905119901119905119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119866T119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905]

(69)

The definition of the Hamiltonian 119867 and (64) leads to119869(119906

⋆ 120585

⋆)minus119869(119906 120585) ge 0 whichmeans that (119906⋆ 120585⋆) is an optimal

control for the problem (14)

The expression (64) is a sufficient condition of optimalityin integral form We want to rewrite this inequality in asuitable form for applications This is the objective of thefollowing theoremwhich could be seen as a natural extensionof [2 Theorem 22] to the jump setting and [3 Theorem 21]to mixed regular-singular control problems

Theorem 9 (sufficient conditions of optimality) Let (119906⋆ 120585⋆)be an admissible control and 119909⋆ the associated controlled stateprocess Let (119901 119902 119903(sdot)) be the unique solution of 119861119878119863119864 (18) Letone assume that (119909 119906) rarr 119867(119905 119909 119906 119901

119905 119902

119905 119903

119905(sdot)) and 119909 rarr

119892(119909) are concave functions If in addition one assumes that

(i) for all 119905 isin [0 119879] V isin 1198601

119867(119905 119909⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) = sup

Visin1198601

119867(119905 119909⋆

119905 V 119901

119905 119902

119905 119903

119905(sdot))

119889119905mdash119886119890 Pmdash119886119904(70)

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Stochastic Analysis

(ii) for all 119905 isin [0 119879] with probability 1

119896119894

119905+ 119866

119894

119905119901119905le 0 for 119894 = 1 119898 (71)

119898

sum

119894=1

1119896119894

119905+119866119894

119905119901119905le0119889120585

⋆119888119894

119905= 0 (72)

119896119894

119905+ 119866

119894

119905(119901

119905minus+ Δ

119873119901119905) le 0 for 119894 = 1 119898 (73)

119898

sum

119894=1

1119896119894

119905+119866119894

119905(119901119905minus+Δ119873119901119905)le0Δ120585

⋆119894

119905= 0 (74)

Then (119906⋆ 120585⋆) is an optimal control

Proof Using (71) and (72) yields

E [int119879

0

119896119905+ 119866

119905119901119905 119889120585

⋆119888

119905] = E[

119898

sum

119894=1

int

119879

0

119896119894

119905+ 119866

119894

119905119901119905 119889120585

⋆119888119894

119905] = 0

(75)

The same computations applied to (73) and (74) imply

E[ sum0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ120585

⋆

119905] = 0 (76)

Hence from Definition 1 we have the following inequal-ity

E[int119879

0

119896119905+ 119866

119905119901119905 119889(120585 minus 120585

⋆)119888

119905

+ sum

0lt119905le119879

119896119905+ 119866

119905(119901

119905minus+ Δ

119873119901119905) Δ(120585 minus 120585

⋆)119905] le 0

(77)

The desired result follows fromTheorem 8

4 Relation to Dynamic Programming

In this section we come back to the control problem studiedin the previous section We recall a verification theoremwhich is useful to compute optimal controls Then we showthat the adjoint process defined in Section 3 as the uniquesolution to the BSDE (18) can be expressed as the gradientof the value function which solves the HJB variationalinequality

41 A Verification Theorem Let 119909119905119909119904

be the solution of thecontrolled SDE (8) for 119904 ge 119905 with initial value 119909

119905= 119909 To put

the problem in a Markovian framework so that we can applydynamic programming we define the performance criterion

119869(119906120585)

(119905 119909)

= E [int119879

119905

119891 (119904 119909119904 119906

119904) 119889119904 + int

119879

119905

119896119904119889120585

119904+ 119892 (119909

119879) | 119909

119905= 119909]

(78)

Since our objective is to maximize this functional thevalue function of the singular control problem becomes

119881 (119905 119909) = sup(119906120585)isinU

119869(119906120585)

(119905 119909) (79)

If we do not apply any singular control then the infinites-imal generator A119906 associated with (8) acting on func-tions 120593 coincides on 1198622

119887(R119899R) with the parabolic integro-

differential operatorA119906 given by

A119906120593 (119905 119909) =

119899

sum

119894=1

119887119894(119905 119909 119906)

120597120593

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972120593

120597119909119894120597119909119895(119905 119909)

+ int119864

120593 (119905 119909 + 120574 (119905 119909 119906 119890)) minus 120593 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597120593

120597119909119894(119905 119909) ] (119889119890)

(80)

where 119886119894119895 = sum119889

ℎ=1(120590

119894ℎ120590119895ℎ) denotes the generic term of the

symmetric matrix 120590120590119879 The variational inequality associatedto the singular control problem is

max sup119906

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) 119897 = 1 119898 = 0

(81)

for (119905 119909) isin [0 119879] times 119874

119882(119879 119909) = 119892 (119909) forall119909 isin 119874 (82)

1198671and119867119897

2 for 119897 = 1 119898 are given by

1198671(119905 119909 (119882 120597

119905119882119882

119909119882

119909119909) (119905 119909) 119906)

=120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906)

119867119897

2(119905 119909119882

119909(119905 119909)) =

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905

(83)

We start with the definition of classical solutions of thevariational inequality (81)

Definition 10 Let one consider a function119882 isin 11986212([0 119879] times

119874) and define the nonintervention region by

119862 (119882) = (119905 119909) isin [0 119879] times 119874

max1le119897le119898

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905 lt 0

(84)

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 11

We say that119882 is a classical solution of (81) if

120597119882

120597119905(119905 119909) + sup

119906

A119906119882(119905 119909) + 119891 (119905 119909 119906) = 0

forall (119905 119909) isin 119862 (119882)

(85)

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119905le 0

forall (119905 119909) isin [0 119879] times 119874 for 119897 = 1 119898

(86)

120597119882

120597119905(119905 119909) +A

119906119882(119905 119909) + 119891 (119905 119909 119906) le 0

for every (119905 119909 119906) isin [0 119879] times 119874 times 1198601

(87)

The following verification theorem is very useful tocompute explicitly the value function and the optimal controlat least in the case where the value function is sufficientlysmooth

Theorem 11 Let 119882 be a classical solution of (81) with theterminal condition (82) such that for some constants 119888

1ge

1 1198882isin (0infin) |119882(119905 119909)| le 119888

2(1 + |119909|

1198881) Then for all (119905 119909) isin

[0 119879] times 119874 and (119906 120585) isin U

119882(119905 119909) ge 119869(119906120585)

(119905 119909) (88)

Furthermore if there exists (119906⋆ 120585⋆) isin U such that withprobability 1

(119905 119909⋆

119905) isin 119862 (119882) 119871119890119887119890119904119892119906119890 119886119897119898119900119904119905 119890119907119890119903119910 119905 le 119879 (89)

119906⋆

119905isin arg max

119906

A119906119882(119905 119909

⋆

119905) + 119891 (119905 119909

⋆

119905 119906) (90)

119898

sum

119897=1

119899

sum

119894minus1

120597119882

120597119909119894(119905 119909

⋆

119905) 119866

119894119897

119905= 119896

119897

119905119889120585

⋆119888119897

119905= 0 (91)

Δ120585119882(119905 119909

⋆

119905) +

119898

sum

119897=1

119896119897

119905Δ120585

⋆119897

119905= 0 (92)

for all jumping times 119905 of 120585⋆119905 then it follows that 119882(119905 119909) =

119869(119906⋆

120585⋆

)(119905 119909)

Proof See [8 Theorem 52]

In the following we present an example on optimalharvesting from a geometric Brownian motion with jumpssee for example [5 8]

Example 12 Consider a population having a size 119883 = 119883119905

119905 ge 0which evolves according to the geometric Levy processthat is

119889119883119905= 120583119883

119905119889119905 + 120590119883

119905119889119861

119905

+ 120579119883119905minusintR+

119890 (119889119905 119889119890) minus 119889120585119905 for 119905 isin [0 119879]

1198830minus= 119909 gt 0

(93)

Here 120585119905is the total number of individuals harvested up

to time 119905 If we define the price per unit harvested at time119905 by 119896(119905) = 119890minus120600119905 and the utility rate obtained when the sizeof the population at 119905 is 119883

119905by 119890minus120600119905119883120574

119905 Then the objective is

to maximize the expected total time-discounted value of theharvested individuals startingwith a population of size119909 thatis

119869 (120585) = E [int119879

0

119890minus120600119905119883

120574

119905119889119905 + int

[0119879)

119890minus120600119905119889120585

119905] (94)

where 119879 = inf119905 ge 0 119883119905= 0 is the time of complete

depletion 120574 isin (0 1) and 120583 120590 120600 120579 are positive constants with12059022 + 120579 int

R+

119890](119889119890) le 120583 lt 120600 The harvesting admissiblestrategy 120585

119905is assumed to be nonnegative nondecreasing

continuous on the right satisfying 119864|120585119879|2lt infin with 120585

0minus= 0

and such that 119883119905gt 0 We denote by Π(119909) the class of such

strategies For any 120585 define

120601 (119905 119909) = sup120585isinΠ(119905119909)

119869120585(119905 119909) (95)

Note that the definition of Π(119905 119909) is similar to Π(119909) exceptthat the starting time is 119905 and the state at 119905 is 119909

If we guess the nonintervention region119862has the form119862 =(119905 119909) 0 lt 119909 lt 119887 for some barrier point 119887 gt 0 then (84)gets the form

0 =120597Φ

120597119905(119905 119909) + 120583119909

120597Φ

120597119909(119905 119909) +

1

212059021199092 120597

2Φ

1205971199092(119905 119909)

+ intR+

Φ (119905 119909 (1 + 120579119890)) minus Φ (119905 119909) minus 120579119909119890120597Φ

120597119909(119905 119909) ] (119889119890)

+ 119909120574 exp (minus120600119905)

(96)

for 0 lt 119909 lt 119887 We try a solutionΦ of the form

Φ (119905 119909) = Ψ (119909) exp (minus120600119905) (97)

hence

AΦ (119905 119909) = exp (minus120600119905)A0Ψ (119909) (98)

whereΨ is the fundamental solution of the ordinary integro-differential equation

minus 120600Ψ (119909) + 120583119909Ψ1015840(119909) +

1

212059021199092Ψ

10158401015840(119909)

+ intR+

Ψ (119909 (1 + 120579119890)) minus Ψ (119909) minus 120579119909119890Ψ1015840(119909) ] (119889119890)

+ 119909120574= 0

(99)

Wenotice thatΨ(119909) = 119860119909120588+119870119909120574 for some arbitrary constant119860 we get

AΦ (119905 119909) = 119909120574(119860ℎ

1(120588) + ℎ

2(120574)) exp (minus120600119905) (100)

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Journal of Stochastic Analysis

where

ℎ1(120588) =

1

212059021205882+ (120583 minus

1

21205902) 120588

+ intR+

(1 + 120579119890)120588minus 1 minus 120579119890120588 ] (119889119890) minus 120600

ℎ2(120574) = 119870(

1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600) +1

(101)

Note that ℎ1(1) = 120583minus120600 lt 0 and lim

119903rarrinfinℎ1(120588) = infin then

there exists 120588 gt 1 such that ℎ1(120588) = 0The constant119870 is given

by

119870 = minus (1

212059021205742+ (120583 minus

1

21205902) 120574

+intR+

(1 + 120579119890)120574minus 1 minus 120579119890120574 ] (119889119890) minus 120600)

minus1

(102)

Outside 119862 we require that Ψ(119909) = 119909 + 119861 where 119861 is aconstant to be determined This suggests that the value mustbe of the form

Φ (119905 119909) = (119860119909

120588+ 119870119909

120574) exp (minus120600119905) for 0 lt 119909 lt 119887

(119909 + 119861) exp (minus120600119905) for 119909 ge 119887(103)

Assuming smooth fit principle at point 119887 then the reflec-tion threshold is

119887 = (119870120574 (1 minus 120574)

119860120588 (120588 minus 1))

1(120588minus120574)

(104)

where

119860 =1 minus 119870120574119887

120574minus1

120588119887120588minus1

119861 = 119860119887120588+ 119870119887

120574minus 119887

(105)

Since 120574 lt 1 and 120588 gt 1 we deduce that 119887 gt 0To construct the optimal control 120585⋆ we consider the

stochastic differential equation

119889119883⋆

119905= 120583119883

⋆

119905119889119905 + 120590119883

⋆

119905119889119861

119905+ int

R+

120579119883⋆

119905119890 (119889119905 119889119890) minus 119889120585

⋆

119905

(106)

119883⋆

119905le 119887 119905 ge 0 (107)

1119883⋆

119905lt119887119889120585

⋆119888

119905= 0 (108)

1119883⋆

119905minus+Δ119873119883⋆

119905le119887Δ120585

⋆

119905= 0 (109)

and if this is the case then

Δ120585⋆

119905= min 119897 gt 0 119883⋆

119905minus+ Δ

119873119883

⋆

119905minus 119897 = 119887 (110)

Arguing as in [7] we can adapt Theorem 15 in [16] to obtainan identification of the optimal harvesting strategy as a localtime of a reflected jump diffusion process Then the system(106)ndash(109) defines the so-called Skorokhod problem whosesolution is a pair (119883⋆

119905 120585

⋆

119905) where 119883⋆

119905is a jump diffusion

process reflected at 119887The conditions (89)ndash(92) ensure the existence of an

increasing process 120585⋆119905such that 119883⋆

119905stays in 119862 for all times

119905 If the initial size 119909 le 119887 then 120585⋆119905is nondecreasing and his

continuous part 120585⋆119888119905

increases only when 119883⋆

119905= 119887 so as to

ensure that119883⋆

119905le 119887

On the other hand we only have Δ120585⋆119905gt 0 if the initial

size 119909 gt 119887 then 120585⋆0minus= 119909 minus 119887 or if 119883⋆

119905jumps out of the

nonintervention region by the random measure 119873 that is119883

⋆

119905minus+ Δ

119873119883

⋆

119905gt 119887 In these cases we get Δ120585⋆

119905gt 0 immediately

to bring119883⋆

119905to 119887

It is easy to verify that if (119883⋆ 120585

⋆) is a solution of the

Skorokhod problem (106)ndash(109) then (119883⋆ 120585

⋆) is an optimal

solution of the problem (93) and (94)By the construction of 120585⋆ andΦ all the conditions of the

verificationTheorem 11 are satisfiedMore precisely the valuefunction along the optimal state reads as

Φ(119905 119883⋆

119905) = (119860119883

⋆120588

119905+ 119870119883

⋆120574

119905) exp (minus120600119905)

for all 119905 isin [0 119879] (111)

42 Link between the SMP and DPP Compared with thestochastic maximum principle one would expect that thesolution (119901 119902 119903(sdot)) of BSDE (18) to correspond to the deriva-tives of the classical solution of the variational inequalities(81)-(82) This is given by the following theorem whichextends [3 Theorem 31] to control problems with a singularcomponent and [2 Theorem 33] to diffusions with jumps

Theorem 13 Let 119882 be a classical solution of (81) with theterminal condition (82) Assume that 119882 isin 119862

13([0 119879] times 119874)

with 119874 = R119899 and there exists (119906⋆ 120585⋆) isin U such that theconditions (89)ndash(92) are satisfied Then the solution of theBSDE (18) is given by

119901119905= 119882

119909(119905 119909

⋆

119905)

119902119905= 119882

119909119909(119905 119909

⋆

119905) 120590 (119905 119909

⋆

119905 119906

⋆

119905)

119903119905(sdot) = 119882

119909(119905 119909

⋆

119905+ 120574 (119905 119909

⋆

119905 119906

⋆

119905 119890)) minus 119882

119909(119905 119909

⋆

119905)

(112)

Proof Throughout the proof we will use the followingabbreviations for 119894 119895 = 1 119899 and ℎ = 1 119889

1206011(119905) = 120601

1(119905 119909

⋆

119905 119906

⋆

119905)

for 1206011= 119887

119894 120590

119894 120590

119894ℎ 120590 119886

119894119895120597119887

119894

120597119909119896120597119887

120597119909119896120597119886

119894119895

120597119909119896120597120590

119894ℎ

120597119909119896120597119891

120597119909119896

1206012(119905 119890) = 120601

2(119905 119909

⋆

119905 119906

⋆

119905 119890) for 120601

2= 120574 120574

119894120597120574

119894

120597119909119896120597120574

120597119909119896

120574minus(119905 119890) = 120574 (119905 119909

⋆

119905minus u⋆

119905 119890) 120574

119894

minus(119905 119890) = 120574

119894(119905 119909

⋆

119905minus 119906

⋆

119905 119890)

(113)

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 13

From Itorsquos rule applied to the semimartingale (120597119882120597119909

119896)(119905 119909

⋆

119905) one has

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905) + int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 119889119909

⋆119894

119904

+1

2int

120591⋆

119877

119905

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904) 119889119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) 120574

119894

minus(119904 119890)119873 (119889119904 119889119890)

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904minus) Δ

120585119909⋆119894

119904

(114)

where 120591⋆ is defined as in Theorem 11 and the sum is takenover all jumping times 119904 of 120585⋆ Note that

Δ120585119909⋆119894

119904= 119909

⋆119894

119904minus (119909

⋆119894

119904minus+ Δ

119873119909⋆119894

119904) =

119898

sum

119897=1

119866119894119897

119904Δ120585

⋆119897

119904

for 119894 = 1 119899

(115)

where Δ120585⋆119897119904= 120585

⋆119897

119904minus 120585

⋆119897

119904minusis a pure jump process Then we can

rewrite (114) as follows

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

)

=120597119882

120597119909119896(119905 119909

⋆

119905)

+ int

120591⋆

119877

119905

1205972119882

120597119904120597119909119896(119904 119909

⋆

119904) +

119899

sum

119894=1

119887119894(119904)

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

+1

2

119899

sum

119894119895=1

119886119894119895(119904)

1205973119882

120597119909119896120597119909119894120597119909119895(119904 119909

⋆

119904)

+ int119864

(120597119882

120597119909119896(119904 119909

⋆

119904+ 120574 (119904 119890)) minus

120597119882

120597119909119896(119904 119909

⋆

119904minus)

minus

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120574

119894(119904 119890)) ] (119889119890) 119889119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 120590

119894(119904) 119889119861

119904

+ int

120591⋆

119877

119905

int119864

120597119882

120597119909119896(119904 119909

⋆

119904minus+ 120574

minus(119904 119890))

minus120597119882

120597119909119896(119904 119909

⋆

119904minus) (119889119904 119889119890)

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904)

119898

sum

119897=1

119866119894119897

119904119889120585

⋆119888119897

119904

+ sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904)

(116)

Let 120585⋆119888119904

denotes the continuous part of 120585⋆119904 that is 120585⋆119888

119904= 120585

⋆

119904minus

sum119905lt119904le120591

⋆

119877

Δ120585⋆119897

119904 Then we can easily show that

int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

119904119889120585

⋆119888119897

119904

= int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904

+ int

120591⋆

119877

119905

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904

(117)

For every (119905 119909) isin 119863119897 using (88) we have

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909) 119866

119894119897

119905=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119905 119909) 119866

119894119897

119905+ 119896

119897

119904 = 0

for 119897 = 1 119898(118)

This proves

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119863119897119889120585

⋆119888119897

119904= 0 (119)

Furthermore for every (119905 119909) isin 119862119897and 119897 = 1 119898 we have

sum119899

119894=1(120597119882120597119909

119896120597119909

119894)(119905 119909)119866

119894119897

119905lt 0

But (91) implies that sum119898

119897=11(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 thus

int

120591⋆

119877

119905

119899

sum

119894=1

119898

sum

119897=1

1205972119882

120597119909119896120597119909119894(119904 119909

⋆

119904) 119866

119894119897

1199041(119904119909⋆

119904)isin119862119897119889120585

⋆119888119897

119904= 0 (120)

The mean value theorem yields

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = (

120597119882

120597119909119896)

119909

(119904 119910 (119904)) Δ120585119909⋆

119904 (121)

where 119910(119904) is some point on the straight line between 119909⋆119904minus+

Δ119873119909⋆

119904and 119909⋆

119904 and (120597119882120597119909119896)

119909represents the gradient matrix

of 120597119882120597119909119896 To prove that the right-hand side of the above

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 International Journal of Stochastic Analysis

equality vanishes it is enough to check that if Δ120585⋆119897119904gt 0 then

sum119899

119894=1(120597

2119882120597119909

119896120597119909

119894)(119904 119910(119904))119866

119894119897

119904= 0 for 119897 = 1 119898 It is clear

by (92) that

0 = Δ120585119882(119904 119909

⋆

119904) +

119898

sum

119897=1

119896119897

119904Δ120585

⋆119897

119904

=

119898

sum

119897=1

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904Δ120585

⋆119897

119904

(122)

Since Δ120585⋆119897119904gt 0 then (119904 119910(119904)) isin 119863

119897 for 119897 = 1 119898

According to (88) we obtain

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119904 119910 (119904)) 119866

119894119897

119904

=120597

120597119909119896

119899

sum

119894=1

120597119882

120597119909119894(119904 119910 (119904)) 119866

119894119897

119904+ 119896

119897

119904 = 0

(123)

This shows that

sum

119905lt119904le120591⋆

119877

Δ120585

120597119882

120597119909119896(119904 119909

⋆

119904) = 0 (124)

On the other hand define

119860 (119905 119909 119906) =120597119882

120597119905(119905 119909) +

119899

sum

119894=1

119887119894(119905 119909 119906)

120597119882

120597119909119894(119905 119909)

+1

2

119899

sum

119894119895=1

119886119894119895(119905 119909 119906)

1205972119882

120597119909119894120597119909119895(119905 119909) + 119891 (119905 119909 119906)

+ int119864

119882(119905 119909 + 120574 (119905 119909 119906 119890)) minus 119882 (119905 119909)

minus

119899

sum

119894=1

120574119894(119905 119909 119906 119890)

120597119882

120597119909119894(119905 119909) ] (119889119890)

(125)

If we differentiate 119860(119905 119909 119906) with respect to 119909119896 andevaluate the result at (119909 119906) = (119909⋆

119905 119906

⋆

119905) we deduce easily from

(84) (89) and (90) that

1205972119882

120597119905120597119909119896(119905 119909

⋆

119905) +

119899

sum

119894=1

119887119894(119905)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

119886119894119895(119905)

1205973119882

120597119909119896120597119909119894120597119909119895(119905 119909

⋆

119905)

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905)

minus

119899

sum

119894=1

120574119894(119904 119890)

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) ] (119889119890)

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

minus1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905 119909

⋆

119905)1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) minus

120597119891

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905)

minus int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times 120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905) ] (119889119890)

(126)

Finally substituting (119) (120) (124) and (126) into (116)yields

119889(120597119882

120597119909119896(119905 119909

⋆

119905))

= minus

119899

sum

119894=1

120597119887119894

120597119909119896(119905)120597119882

120597119909119894(119905 119909

⋆

119905)

+1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905) +

120597119891

120597119909119896(119905)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119890)

times(120597119882

120597119909119894(119905 119909

⋆

119905+ 120574 (119905 119890)) minus

120597119882

120597119909119894(119905 119909

⋆

119905))] (119889119890)119889119905

+

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894(119905) 119889119861

119905

+ int119864

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574

minus(119905 119890))minus

120597119882

120597119909119896(119905 119909

⋆

119905minus) (119889119905 119889119890)

(127)

The continuity of 120597119882120597119909119896 leads to

lim119877rarrinfin

120597119882

120597119909119896(120591

⋆

119877 119909

⋆

120591⋆

119877

) =120597119882

120597119909119896(119879 119909

⋆

119879)

=120597119892

120597119909119896(119909

⋆

119879) for each 119896 = 1 119899

(128)

Clearly

1

2

119899

sum

119894119895=1

120597119886119894119895

120597119909119896(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=1

2

119899

sum

119894119895=1

120597

120597119909119896(

119889

sum

ℎ=1

120590119894ℎ(119905) 120590

119895ℎ(119905))

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

119905)

=

119899

sum

119895=1

119889

sum

ℎ=1

(

119899

sum

119894=1

120590119894ℎ(119905)

1205972119882

120597119909119894120597119909119895(119905 119909

⋆

t ))120597120590

119894ℎ

120597119909119896(119905)

(129)

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 15

Now from (17) we have

120597119867

120597119909119896(119905 119909 119906 119901 119902 119903 (sdot))

=

119899

sum

119894=1

120597119887119894

120597119909119896(119905 119909 119906) 119901

119894

+

119889

sum

ℎ=1

119899

sum

119894=1

120597120590119894ℎ

120597119909119896(119905 119909 119906) 119902

119894ℎ+120597119891

120597119909119896(119905 119909 119906)

+ int119864

119899

sum

119894=1

120597120574119894

120597119909119896(119905 119909 119906 119890) 119903

119894(119890) ] (119889119890)

(130)

The 119896th coordinate 119901119896119905of the adjoint process 119901

119905satisfies

119889119901119896

119905= minus

120597119867

120597119909119896(119905 119909

⋆

119905 119906

⋆

119905 119901

119905 119902

119905 119903

119905(sdot)) 119889119905

+ 119902119896

119905119889119861

119905+ int

119864

119903119896

119905minus(119890) (119889119905 119889119890) for 119905 isin [0 119879]

119901119896

119879=120597119892

120597119909119896(119909

⋆

119879)

(131)

with 119902119896119905119889119861

119905= sum

119889

ℎ=1119902119896ℎ

119905119889119861

ℎ

119905 Hence the uniqueness of the

solution of (131) and relation (128) allows us to get

119901119896

119905=120597119882

120597119909119896(119905 119909

⋆

119905)

119902119896ℎ

119905=

119899

sum

119894=1

1205972119882

120597119909119896120597119909119894(119905 119909

⋆

119905) 120590

119894ℎ(119905)

119903119896

119905minus(sdot) =

120597119882

120597119909119896(119905 119909

⋆

119905minus+ 120574 (119905 119890)) minus

120597119882

120597119909119896(119905 119909

⋆

119905minus)

(132)

where 119902119896ℎ119905is the generic element of the matrix 119902

119905and 119909⋆

119905is the

optimal solution of the controlled SDE (8)

Example 14 We return to the same example in the previoussection

Now we illustrate a verification result for the maximumprinciple We suppose that 119879 is a fixed time In this case theHamiltonian gets the form

119867(119905 119883119905 119901

119905 119902

119905 119903

119905(sdot)) = 120583119883

119905119901119905+ 120590119883

119905119902119905+ 119883

120574

119905(minus120600119905)

+ 120579119883119905minusintR+

119890119903119905(119890) ] (119889119890)

(133)

Let 120585⋆ be a candidate for an optimal control and let119883⋆ bethe corresponding state process with corresponding solution

(119901⋆ 119902

⋆ 119903

⋆(sdot)) of the following adjoint equation for all 119905 isin

[0 119879)

119889119901⋆

119905= minus (120583119901

⋆

119905+ 120590119902

⋆

119905+ 120579int

R+

119890119903⋆

119905(119890) ] (119889119890)

+120574119883⋆120574minus1

119905exp (minus120600119904) ) 119889119905

+ 119902⋆

119905119889119861

119905+ int

R+

119903⋆

119905minus(119890) (119889119905 119889119890)

(134)

minus119901⋆

119905+ exp (minus120600119905) le 0 forall119905 (135)

1minus119901⋆

119905+exp(minus120600119905)lt0119889120585

⋆119888

119905= 0 (136)

minus (119901⋆

119905minus+ Δ

119873119901⋆

119905) + exp (minus120600119905) le 0 (137)

1minus(119901⋆

119905minus+Δ119873119901⋆

119905)+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (138)

Since 119892 = 0 we assume the transversality condition

E [119901⋆

119879(119883

⋆

119879minus 119883

119879)] le 0 (139)

We remark that Δ120585119901⋆

119905= 0 then 119901⋆

119905minus+ Δ

119873119901⋆

119905= 119901

⋆

119905 and

the condition (138) reduces to

1minus119901⋆

119905+exp(minus120600119905)lt0Δ120585

⋆

119905= 0 (140)

We use the relation between the value function and thesolution (119901⋆ 119902⋆ 119903⋆(119890)) of the adjoint equation along theoptimal state We prove that the solution of the adjointequation is represented as

119901⋆

119905= (119860120588119883

⋆120588minus1

119905+ 119870120574119883

⋆120574minus1

119905) exp (minus120600119905)

119902⋆

119905= 120590 (119860120588 (120588 minus 1)119883

⋆120588minus1

119905+ 119870120574 (120574 minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

119903⋆

119905(119890) = (119860120588 ((1 + 120579119890)

120588minus1minus 1)119883

⋆120588minus1

119905

+119870120574 ((1 + 120579119890)120574minus1minus 1)119883

⋆120574minus1

119905) exp (minus120600119905)

(141)

for all 119905 isin [0 119879)To see this we differentiate the process (119860120588119883⋆120588minus1

119905+

119870120574119883⋆120574minus1

119905) exp(minus120600119905) using Itorsquos rule for semimartingales and

by using the same procedure as in the proof of Theorem 13Then the conclusion follows readily from the verification of(135) (136) and (139) First an explicit formula for119883

119905is given

in [4] by

119883119905= 119890

120583119905119872

119905119909 minus (int

[0119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

for 119905 isin [0 119879]

(142)

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 International Journal of Stochastic Analysis

where 120573119905= (int

R+

120579119890119873(119905 119889119890))(1 + intR+

120579119890119873(119905 119889119890))minus1 and

119872119905is a geometric Levy process defined by

119872119905= exp (minus1

21205902+ int

R+

ln (1 + 120579119890) minus 120579119890 ] (119889119890)) 119905

+ 120590119861119905+ int

119905

0

intR+

ln (1 + 120579119890) (119889119905 119889119890) (143)

From the representation (142) and by the fact that119883⋆

119879and119905le

119909119872119879and119905

exp(120583(119879 and 119905)) we get

1 minus119883

119879and119905

119883⋆

119879and119905

le1

119909(int

[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904) lt infin

(144)

henceE [119901

⋆

119879and119905(119883

⋆

119879and119905minus 119883

119879and119905)]

le E[((119860120588119883⋆120588

119879and119905+ 119870120574119883

⋆120574

119879and119905) exp (minus120600 (119879 and 119905)))2]

12

times E[

[

(1

119909int[0119879and119905)

119872minus1

119904exp (minus120583119904) 119889120585

119904

+ sum

0lt119904le119879and119905

119872minus1

119904120573119904exp (minus120583119904) Δ120585

119904)

2

]

]

12

(145)By the dominated convergence theorem we obtain (139)

by sending 119905 to infinity in (145)A simple computation shows that the conditions (135)ndash

(138) are consequences of (107)ndash(109) This shows in partic-ular that the pair (119883⋆

119905 120585

⋆

119905) satisfies the optimality sufficient

conditions and then it is optimal This completes the proofof the following result

Theorem 15 One supposes that 12059022 + 120579 intR+

119890](119889119890) le 120583 lt 120600and 119890 ge 0 119889] minus 119886119890 If the strategy 120585⋆ is chosen such that thecorresponding solution of the adjoint process is given by (141)then this choice is optimal

Remark 16 In this example it is shown in particular that therelationship between the stochastic maximum principle anddynamic programming could be very useful to solve explicitlyconstrained backward stochastic differential equations withtransversality condition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the referees and the associateeditor for valuable suggestions that led to a substancial

improvement of the paper This work has been partiallysupported by the Direction Generale de la Recherche Sci-entifique et du Developpement Technologique (Algeria)under Contract no 051PNRUMKB2011 and by the FrenchAlgerian Cooperation Program Tassili 13 MDU 887

References

[1] S Bahlali and A Chala ldquoThe stochastic maximum principlein optimal control of singular diffusions with non linearcoefficientsrdquo Random Operators and Stochastic Equations vol13 no 1 pp 1ndash10 2005

[2] K Bahlali F Chighoub and B Mezerdi ldquoOn the relationshipbetween the stochastic maximum principle and dynamic pro-gramming in singular stochastic controlrdquo Stochastics vol 84no 2-3 pp 233ndash249 2012

[3] N C Framstad B Oslashksendal and A Sulem ldquoSufficient stochas-tic maximum principle for the optimal control of jump dif-fusions and applications to financerdquo Journal of OptimizationTheory and Applications vol 121 pp 77ndash98 2004 Erratum inJournal of OptimizationTheory and Applications vol 124 no 2pp 511ndash512 2005

[4] B Oslashksendal and A Sulem ldquoSingular stochastic control andoptimal stopping with partial information of Ito-Levy pro-cessesrdquo SIAM Journal on Control and Optimization vol 50 no4 pp 2254ndash2287 2012

[5] L H R Alvarez and T A Rakkolainen ldquoOn singular stochasticcontrol and optimal stopping of spectrally negative jumpdiffusionsrdquo Stochastics vol 81 no 1 pp 55ndash78 2009

[6] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USA1993

[7] N C Framstad BOslashksendal andA Sulem ldquoOptimal consump-tion and portfolio in a jump diffusionmarket with proportionaltransaction costsrdquo Journal of Mathematical Economics vol 35no 2 pp 233ndash257 2001 Arbitrage and control problems infinance

[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2005

[9] U G Haussmann and W Suo ldquoSingular optimal stochasticcontrols II Dynamic programmingrdquo SIAM Journal on Controland Optimization vol 33 no 3 pp 937ndash959 1995

[10] J A Bather andH Chernoff ldquoSequential decision in the controlof a spaceship (finite fuel)rdquo Journal of Applied Probability vol49 pp 584ndash604 1967

[11] V E Benes L A Shepp and H S Witsenhausen ldquoSomesolvable stochastic control problemsrdquo Stochastics vol 4 no 1pp 39ndash83 198081

[12] P-L Lions and A-S Sznitman ldquoStochastic differential equa-tions with reflecting boundary conditionsrdquo Communications onPure and Applied Mathematics vol 37 no 4 pp 511ndash537 1984

[13] F M Baldursson and I Karatzas ldquoIrreversible investment andindustry equilibriumrdquo Finance and Stochastics vol 1 pp 69ndash891997

[14] E M Lungu and B Oslashksendal ldquoOptimal harvesting from apopulation in a stochastic crowded environmentrdquoMathematicalBiosciences vol 145 no 1 pp 47ndash75 1997

[15] M H A Davis and A R Norman ldquoPortfolio selection withtransaction costsrdquo Mathematics of Operations Research vol 15no 4 pp 676ndash713 1990

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 17

[16] M Chaleyat-Maurel N El Karoui and B Marchal ldquoReflexiondiscontinue et systemes stochastiquesrdquoThe Annals of Probabil-ity vol 8 no 6 pp 1049ndash1067 1980

[17] P L Chow J-L Menaldi and M Robin ldquoAdditive control ofstochastic linear systems with finite horizonrdquo SIAM Journal onControl and Optimization vol 23 no 6 pp 858ndash899 1985

[18] A Cadenillas and U G Haussmann ldquoThe stochastic maxi-mum principle for a singular control problemrdquo Stochastics andStochastics Reports vol 49 no 3-4 pp 211ndash237 1994

[19] S Bahlali and B Mezerdi ldquoA general stochastic maximumprinciple for singular control problemsrdquo Electronic Journal ofProbability vol 10 pp 988ndash1004 2005

[20] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[21] S Bahlali B Djehiche and B Mezerdi ldquoThe relaxed stochasticmaximum principle in singular optimal control of diffusionsrdquoSIAM Journal on Control and Optimization vol 46 no 2 pp427ndash444 2007

[22] K Bahlali F Chighoub B Djehiche and BMezerdi ldquoOptimal-ity necessary conditions in singular stochastic control problemswith nonsmooth datardquo Journal of Mathematical Analysis andApplications vol 355 no 2 pp 479ndash494 2009

[23] H Pham ldquoOptimal stopping of controlled jump diffusion pro-cesses a viscosity solution approachrdquo Journal of MathematicalSystems Estimation and Control vol 8 pp 1ndash27 1998

[24] J-T Shi and Z Wu ldquoRelationship betweenMP and DPP for thestochastic optimal control problem of jump diffusionsrdquoAppliedMathematics and Optimization vol 63 no 2 pp 151ndash189 2011

[25] S J Tang and X J Li ldquoNecessary conditions for optimal controlof stochastic systems with random jumpsrdquo SIAM Journal onControl and Optimization vol 32 no 5 pp 1447ndash1475 1994

[26] X Y Zhou ldquoA unified treatment of maximum principle anddynamic programming in stochastic controlsrdquo Stochastics andStochastics Reports vol 36 no 3-4 pp 137ndash161 1991

[27] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[28] A Eyraud-Loisel ldquoBackward stochastic differential equationswith enlarged filtration option hedging of an insider trader ina financial market with jumpsrdquo Stochastic Processes and TheirApplications vol 115 no 11 pp 1745ndash1763 2005

[29] G Barles R Buckdahn and E Pardoux ldquoBSDEs andintegral-partial differential equationsrdquo Stochastics and Stochas-tics Reports vol 60 no 1-2 pp 57ndash83 1997

[30] N Ikeda and S Watanabe Stochastic Differential Equations andDiffusion Processes vol 24 North-Holland Amsterdam TheNetherlands 2nd edition 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of