EXISTENCE AND NECESSARY CONDITIONS IN THE CONTROL OF … · EXISTENCE AND NECESSARY CONDITIONS IN...

SUMMARYFormulation of the problem

The relaxed control problemExistence

The maximum principle

EXISTENCE AND NECESSARY CONDITIONSIN THE CONTROL OF FBSDE

Brahim Mezerdi

Laboratory of Applied Mathematics

University of Biskra, Algeria

Workshop on Finance and Insurance

Jena, March 16-20, 2008

Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE




SUMMARY

Joint work with K. Bahlali (Univ. Toulon France) and B. Gherbal (Univ.Biskra)

I Formulation of the problem

I The Relaxed Problem

I Existence result

I Maximum principle





Formulation of the problem

The systems we wish to control are driven by the following d-dimensionalFBSDE

dXt = b (t,Xt ,Ut) dt + σ (t,Xt) dWt ,X (0) = x ,−dYt = f (t,Xt ,Yt ,Ut) dt − ZtdMX

t ,Y (T ) = g (XT ) .

(1.1)






b, σ, f and g are given, MX is the martingale part of the diffusionprocess X , (Wt , t ≥ 0) is a standard Brownian motion, defined on(Ω,F , (Ft)t≥0 ,P

),. The control variable ut , called strict control, is an

Ft adapted process with values in some compact metric space A.The expected cost on the time interval [0,T ] is of the form

J (U) = E

[l (Y0) +

∫ T

0

h (t,Xt ,Yt ,Ut) dt

]. (1.2)






Let (Wt , t ≥ 0) be a d-dimensional Brownian motion defined on some

filtered probability space(Ω,F , (Ft)t≥0 ,P

).

(A1) Assume that

b : [0,T ]× Rd × A → Rd ,

σ : [0,T ]× Rd → Rd×d ,

f : [0,T ]× Rd × Rd × A → Rk

g : [0,T ]× Rd → Rk

are bounded, measurable and continuous.We assume also that f is uniformly Lipschitz in (x , y) , uniformly withrespect to u.






We can reformulate the above control problem as an equivalentmartingale problem. This simplifies taking limits. Let L be theinfinitesimal generator, associated with (2.1), acting on functions ϕ inC 2

b

(Rd ; R

)defined by

Lϕ (t, x , u) :=1

2

∑i,j

(aij

∂2ϕ

∂xi∂xj

)(t, x) +

∑i

(bi∂ϕ

∂xi

)(t, x , u) , (2.2)

where aij (t, x , u) denotes the generic term of the symmetric matrixσσ∗ (t, x , u).






Definition 2.1. A strict control is a collectionU = (Ω,F , Ftt ,P,Zt ,Xt , x ,Yt ,Ut) such that(1) x ∈ Rd is the initial data,

(2)(Ω,F , Ftt10 ,P

)is a filtered probability space satisfying the

usual conditions,(3) Ut is a A-valued process, Ft-progressively measurable,






(4) (Xt)t is an Rd -valued, Ft-adapted, with continuous paths, such thatX (0) = x and satisfies for each ϕ ∈ C 2

b

(Rd ,R

),

ϕ (Xt)− ϕ (x)−∫ t

0

Lϕ (s,Xs ,Ut) ds is a P-martingale, (2.3)

where L is the infinitesimal generator defined by (2.2),(5) (Yt)t is Rd -valued, Ft-adapted, such that

Yt = g (XT ) +

∫ T

t

f (s,Xs ,Ys ,Ut) ds −∫ T

t

ZsdMXs , 0 ≤ t ≤ T , (2.4)

(6) MXt is the martingale part of the diffusion process X and f (., y , u) is

F t-progressively measurable.






We denote by U the set of strict controls.The cost corresponding to a control U is defined by

J (U) = E

(l (Y0) +

∫ T

0

h (t,Xt ,Yt ,Ut) dt

)(2.5)

whereA3)

h : [0,T ]× Rd × Rd × A → Rk

l : Rd → R

are bounded and continuous functions.





The relaxed control problem

The idea of relaxed control is to replace the A-valued process (Ut) withP (A)-valued process (qt), where P (A) is the space of probabilitymeasures equipped with the topology of weak convergence. Let V be theset of probability measures on [0,T ]× A whose projections on [0,T ]coincide with the Lebesgue measure dt. Equipped with the topology ofstable convergence of measures, V is a compact metrizable space, seeJacod & Memin.






Definition 3.1. A relaxed control is a collectionq :=

(Ω, F , F tt10 ,P,Zt ,Xt , x , ,Yt , qt

)such that

(1) x ∈ Rd is the initial data,

(2)(Ω,F , Ftt10 ,P

)is a filtered probability space satisfying the

usual conditions,(3) (qt)t is a P (A)-valued process, Ft-progressively measurable suchthat for each t, 1(0,t] · q is Ft-measurable.






(4) (Xt)t is an Rd -valued, adapted, with continuous paths, satisfyingX (0) = x and for each ϕ ∈ C 2

b

(Rd ,R

),

ϕ (Xt)− ϕ (x)−∫ t

0

∫A

Lϕ (s,Xs , a) qs (da) ds is a P-martingale. (3.1)

(5) (Yt)t is Rd -valued, Ft-adapted, such that

Yt = g (XT ) +

∫ T

t

∫A

f (s,Xs ,Ys , a) qs (da) ds −∫ T

t

ZsdMXs . (3.2)

(6) MXt is the martingale part of the forward component X .

We denote by R the collection of all relaxed controls.






The cost function associated to a relaxed control q is defined by

J (q) = E

(l (Y0) +

∫ T

0

∫A

h (t,Xt ,Yt , a) qt (da) dt

)(3.3)





Existence

The main result of this section is given by the following.

Theorem 4.1. Under conditions A1 and A2, there exists a relaxedcontrol q ∈ R such that

J (q) = infµ∈R

J (µ) . (4.0)





Existence

Jakubowsky S-TopologyThe S-topology has been introduced by Jakubowski, as a topologydefined on the Skorokhod space of cadlag functions D

([0,T ] ; Rk

). This

topology is weaker than the Skorokhod topology and the tightnesscriteria are easier to establish. These criteria are the same as the oneused in Meyer & Zheng topology.





Existence

A.1)Write xn →S x0, if for every ε > 0 one can find functions νn,ε ofbound ed variation in [0,T ] , which are ε-uniformly close to xn’s andweakly-* convergent

supt∈[0,T ]

|xn (t)− νn,ε (t)| ≤ ε, n = 0, 1, 2, ...

vn,ε →wν0,ε, as n →∞

A.2) K ⊂ D ([0,T ]) is relatively S-compact if and only if, the followingconditions hold

supK

supt∈[0,T ]

|x (t)| ≤ CK <∞

andsupK

Nη (x) ≤ Cη <∞ η > 0,





Existence

We recall (see Jakubowski ) that for a familly (X n)n of quasi-martingales

on the probability space((

Ω, Ft0≤t≤T ,P))

, the following condition

insures the tightness of the familly (X n)n on the space D([0,T ] ; Rk

)endowed with the S-topology

supn

(sup

0≤t≤TE |X n

t |+ CV 0t (X n)

)<∞,

CV 0t (X ) = supE

(∑i

∣∣E (Xti+1 − Xti ) Fnti

∣∣) ,





Existence

To prove the existence theorem, we need some results on the tightness ofprocesses under consideration.

Let (qn)n≥0 be a minimizing sequence, that is limn→∞

J (qn) = infµ∈R

J (µ) .

Let (X n,Y n,Z n) be the solution of (3.1) and (3.2) corresponding to qn.





Existence

Lemma 4.4. The family of relaxed controls (qn)n≥0 is tight in V.

Proposition 4.5. Let X nt be such that

X nt =

∫ t

0

∫A

b (s,X ns , a) qn

s (da) ds +

∫ t

0

σ (s,X ns ) dWs . (4.6)

Then, X n is tight in the space C([0,T ] ,Rk

)endowed with the

topology of the uniform convergence.





Existence

Proposition 4.2. There exists a positive constant C such that

supn

E(

sup0≤t≤T

|Y nt |

2 +

∫ T

t

|Z ns σ(X n

s )|2ds< C (4.1)

Proposition 4.3. The sequence (Y n,∫ ·0Z n

s dMX n

s ) is tight on the space

D([0,T ] ; Rk

)× D

([0,T ] ; Rk

)endowed with the S-topology.





Existence

Sketch of proofLet (qn)n≥0 be the minimizing sequence

ϕ (X nt )− ϕ (x)−

∫ t

0

∫A

Lϕ (s,X ns , a) qn

s (da) ds is a P-martingale

Y nt = g (X n

T ) +

∫ T

t

∫A

f (s,X ns ,Y

ns , a) qn

s (da) ds + Mnt −Mn

T , (4.7)

where Mnt :=

∫ t

0Z n

s dMX n

s , and

limn→∞

J (qn) = limn→∞

E

[l (Y n

0 ) +

∫ T

0

∫A

h (t,X nt ,Y

nt , a) qn

t (da) dt

]= λ.

(4.8)





Existence

From tightness results, it follows that the sequence of processes

γn = (qn,X n,Y n,Mn) (4.9)

is tight on the space

Γ = V×C([0,T ] ,Rk

)×[D([0,T ] ; Rk

)]2(4.10)

equipped with the product topology of the uniform convergence on thefirst factor, the topology of stable convergence of measures on thesecond factor and the S-topology on the third factor.





Existence

By Jakubowski’s theorem, there exists a probability space(Ω, F , P

), a

sequence γn =(X n, qn, Y n, Mn

)and γ =

(X , q, Y , M

)defined on this

space such that(i) for each n ∈ N, law(γn) = law(γn),(ii) there exists a subsequence (γnk ) of (γn) , still denoted (γn) ,

which converges to γ,P-a.s. on the space Γ,(iii) Y n

t converges to Yt , dt × P − a.s.,

(iv) sup0≤t≤T

∣∣∣X nt − Xt

∣∣∣→ 0, as n →∞, P − a.s.

To conclude we pass to the limit in the martingale problem for theforward component and in the BSDE.





Existence

We consider the following assumption:

(A2) (Roxin’s condition): For every (t, x , y) ∈ [0,T ]× Rn × Rm, the set

(b, f , h) (t, x , y ,A)

:= bi (t, x , u) , fj (t, x , y , u) , h (t, x , y , u) u ∈ A, i = 1, ..., n, j = 1, ...,m ,

is convex and closed in Rn+m+1.

Corollary Suppose that assumptions A1, A2, A3 hold, then the relaxedoptimal control qt has the form of a Dirac measure charging a strictcontrol Ut (i.e; dtqt (da) = dtδUt

(da)).






Throughout this section, the following assumptions will be in force.(A4) b, σ, f , g , h and l are continuous in [0,T ]× Rn × Rm × Rk and

continuously differentiable with respect to x , y .

(A5) The derivatives of b, σ, f , g , h and l with respect to x , y areuniformly bounded,

|ρ(t, x , y , u)| ≤ c for ρ = bx , σx , fx , fy , gx , hx , hy

and|ly (t, x , y , u)| ≤ c (1 + |y |)

(A6) there exists a positive constant c such that for every(t, x , x ′, y , y ′, u)

|bx (t, x , u)− bx (t, x ′, u)|+ |σx (t, x)− σx (t, x ′)| ≤ c |x − x ′|

and|ρ (t, x , y , u)− ρ (t, x ′, y ′, u)| ≤ c (|x − x ′|+ |y − y ′|) ,

for ρ = fx , fy , hx , hy .






The set U of strict controls is embedded into the set R of relaxedcontrols, by the mapping

ψ : u ∈ U 7→ ψ (u) (dt, da) = dtδu(t) (da) ∈ R, (5.1)

Lemma 5.2. (chattering lemma ). Let (µt) be a predictable processwith values in the space of probability measures on A. Then there existsa sequence of predictable processes (Un

t ) with values in A, such that thesequence of random measures

(δUn

t(da) dt

)converges weakly to

µt (da) dt,P-a.s.






Denote by (X nt ,Y

nt ,Z

nt ) the solution of FBSDE which which can be

written in the relaxed form asdX n

t =∫A

b (t,X nt , a)µ

nt (da) dt + σ (t,X n

t ) dWt , X n0 = x ,

−dY nt =

∫A

f (t,X nt ,Y

nt , a)µ

nt (da) dt − Z n

t dMX n

t , Y nT = g (X n

T ) .(5.5)

Where µnt (da) = δUn(t) (da) and MX n

t is the martingale part of thediffusion X n.

Theorem 5.3. Let (Xt ,Yt ,Zt) and (X nt ,Y

nt ,Z

nt ) be the solutions of

((5.2),(5.3)) and (5.5) associated with µ and Un, respectively. Then itfollows that

limn→+∞

E

(sup

0≤t≤T|X n

t − Xt |2)

= 0, (5.6)

and

limn→+∞

E

(|Y n

t − Yt |2 +

∫ T

0

|Z nt σ (t,X n

t )− Ztσ (t,Xt)|2 dt

)= 0. (5.7)






Proposition 5.4. Let J (Un) and J (µ) be the expected costscorresponding respectively to Un and µ, where Un and µ are defined asin the last theorem. Then there exist a subsequence (Unk ) of (Un) , stilldenoted by (Un) such that J (Un) converges to J (µ) .






We know from the last section that an optimal relaxed control exists andthat there is a sequence (un) ∈ U of strict controls such that

dtqnt (da) = dtδun

t(da) →

n→+∞dtqt (da) P-a.s in V,

According to the optimality of q, there exists a sequence (εn) of positivereal numbers with lim

n→+∞εn = 0 such that

J (un) = J (qn) ≤ J (q) + εn, (5.8)

where qn = dtδunt(da) .






To derive necessary conditions for near optimality, we use Ekeland’svariational principle along with an appropriate choice of a metric on thespace U of admissible controls.

Define a metric d on the space U by

d (u, v) := P ⊗ dt

(ω, t) ∈ Ω× [0,T ] ; u (ω, t) 6= v (ω, t), (5.9)

where P ⊗ dt is the product measure of P and the Lebesgue measure.






Let us summarize some of the properties satisfied by d

Lemma i) (U, d) is a complete metric space.ii) For any u, v ∈ U along with the corresponding trajectories(X ,Y ,Z ) , (X ′,Y ′,Z ′), it hold that

E

(sup

t∈[0,T ]

|Xt − X ′t |

2

)≤ C1 (d (u, v))

12 (5.10)

and

E

(sup

t∈[0,T ]

|Yt − Y ′t |

2+

∫ T

0

|Ztσ (t,Xt)− Z ′tσ (t,X ′t )|

2dt

)≤ C1 (d (u, v))

12

(5.11)iii) The cost functional J : (U, d) → R is continuous. More precisely ifu, v are two elements in U then

|J (u)− J (v)| ≤ C (d (u, v))12 . (5.12)






For any u ∈ U, let us denote (X ,Y ,Z ) the corresponding trajectory. Weintroduce the adjoint equations and the Hamiltonian function for thecontrol problem.

dpt = −Hx (t,Xt ,Yt , ut , pt ,Qt , kt) dt + ktdWt

pT = gx (XT ) QT(5.13)

anddQt = Hy (t,Xt ,Yt , ut , pt ,Qt , kt) dt +Hz (t,Xt ,Yt , ut , pt ,Qt , kt) dMX

t

Q0 = ly (Y0)(5.14)

The Hamiltonian function is defined by

H (t, x , y , q, p,Q, k) := h (t, x , y , u) + 〈p, b (t, x , u)〉+ 〈k, σ (t, x)〉+ 〈Q, f (t, x , y , u)〉 . (5.15)






Lemma 5.7. For any u, v ∈ U along with the corresponding trajectories(X ,Y ,Z ) , (X ′,Y ′,Z ′) and the solutions (p,Q, k) , (p′,Q ′, k ′) of thecorresponding adjoint equations, it hold that

E

(sup

t∈[0,T ]

|Qt − Q ′t |

2

)≤ C1 (d (u, v))

12 , (5.17)

and

E

∫ T

0

|pt − p′t |

2+ |kt − k ′t |

2

dt ≤ C1 (d (u, v))12 . (5.18)






A suitable version of Ekeland’s variational principle implies that, given asequence of positive real numbers εn > 0 with lim

n→+∞εn = 0, there exists

an admissible control un such that

J (un) ≤ infu∈U

J (u) + εn

andJ (un) ≤ J (v) + εnd (v , un) ;∀v ∈ U.

Remark 5.8. un which is εn-optimal for the cost J (u) is in fact optimalfor the new perturbed cost J (u) = J (u) + εnd (u, un) .






The next proposition gives necessary conditions for near optimalitysatisfied by the minimizing sequence (un).Proposition Let un be an εn − optimalcontrolthen, for any γ ∈

[0, 1

6

)there exists a constant C2 = C2 (γ) > 0 such that

E

∫ T

0

pns (b (s,X n

s , u)− b (s,X ns , u

ns )) + Qn

s (f (s,X ns ,Y

ns , u)− f (s,X n

s ,Yns , u

ns )) ds

+ E

∫ T

0

(h (s,X ns ,Y

ns , u)− h (s,X n

s ,Yns , u

ns )) ds ≥ −C2ε

γn ,∀u ∈ A.

(5.19)

where (pnt ,Q

nt , k

nt ) are the solutions of the adjoint equations

corresponding to (X n,Y n,Z n, un) .






Let q be a relaxed optimal control,(X , Y , Z

)be the corresponding

trajectory.Define the following adjoint processes dpt = −

∫AHx

(t, Xt , Yt , a, pt , Qt , kt

)qt (da) dt + ktdWt

pT = gx

(XT

)QT

(5.25)

anddQt =

∫AHy qt (da) dt +

∫AHz qt (da) dMX

t

Q0 = ly(Y0

),

(5.26)






Theorem ( The Pontryagin relaxed maximum principle). If(X , Y , Z , q

)denotes an optimal relaxed uplet, then there exists a

Lebesgue negligible subset N such that, for any t not in N,∫A

H(t, Xt , Yt , a, pt , Qt , kt

)qt (da) (5.29)

= supq∈P(A)

∫A

H(t, Xt , Yt , a, pt , Qt , kt

)qt (da) , P-a.s.






The proof of the main result is based on the following stability lemma.

Lemma 5.15. Let pn,Qn, kn (resp. p, Q, k) be defined by (5.13), (5.14)(resp. (5.25), (5.26)), then , we have

i”) limn→+∞

E

(sup

0≤t≤T|pn

t − pt |2 +

∫ T

0

∣∣∣knt − kt

∣∣∣2 dt

)= 0,

ii”) limn→+∞

E

(sup

0≤t≤T

∣∣∣Qnt − Qt

∣∣∣2) = 0,

iii”) limn→+∞

∫ T

0

E (H (s,X nt ,Y

nt , u

nt , p

nt ,Q

nt , k

nt )) dt

=

∫ T

0

E(H(s, Xt , Yt , a, pt , Qt , kt

)qt (da)

)dt.


Date post:	14-Mar-2020
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

EXISTENCE AND NECESSARY CONDITIONS IN THE CONTROL OF … · EXISTENCE AND NECESSARY CONDITIONS IN...

Documents