An extension of stochastic calculus tocertain non-Markovian processes

Nicolas Privault

Equipe d’Analyse et Probabilites, Universite d’Evry-Val d’EssonneBoulevard des Coquibus, 91025 Evry Cedex, France1

e-mail: privault@lami.univ-evry.fr

Abstract

By time changes of Levy processes we construct two operators on Fockspace whose sum is a second quantized operator, and that complement theannihilation and creation operators whose probabilistic interpretations useshifts of trajectories. This results in an analytic construction, for certain non-Markovian processes, of stochastic calculus including Ito differentials, genera-tors and associated integro-differential equations, without using the notion offiltration.

Key words: Anticipating stochastic calculus, Malliavin calculus, Fock space, Levy

processes.

Mathematics Subject Classification (1991): 60G55, 60H07, 60G57, 60G50

1 Introduction

The stochastic calculus of variations on the Wiener space, cf. [12], allows to construct

an anticipating stochastic calculus for Brownian motion via the Skorohod integral,

cf. e.g. [14], [15]. Extensions of this anticipating stochastic calculus in the jump case

have been considered in [6], [16], [19], however they only concern the Poisson process

on the real line, or time-changed Poisson processes, cf. [20]. The regularity of laws

of solutions of stochastic differential equations of jump type have been studied in

[4], [5], [18]. There are well-known links between Fock space and stochastic calculus

that usually involve the annihilation and creation operators and their probabilistic

interpretation by shifts of trajectories of the Wiener and Poisson processes. In this

paper we introduce two operators on Fock space that are interpreted probabilistically

by time changes of the Wiener and Poisson processes. We establish a formula that

1Present address: Centre de Recerca Matematica, Apartat 50, 08193 Bellaterra, Spain.

1

expresses on random variables the infinitesimal action of time changes performed on

a Levy process X(dx, dt) with Brownian component B(dt) and compensated Poisson

component N(dx, dt) on M × IR+, and study its connections to the Ito formula. For

a sufficiently smooth random variable F ,

− d

dεTεhF|ε=0 = ahF +

∫M×IR+

h(x, t)

(1x 6=0∇−x,t +

1

21x=0∇−0,t∇−0,t

)Fµ(dx)dt.

The operator ∇ is defined by means of chaotic decompositions, ∇− is the annihila-

tion operator on Fock space, and ThF is defined by evaluationg F on time changed

trajectories of the Levy process. (This time change is governed by the function

h ∈ L∞(IR+)). The term ahF has expectation is zero and is interpreted as a martin-

gale term. The sum of the operator ∇ and its adjoint ∇⊕ gives a second quantized

operator on Fock space close to the number operator. This can be interpreted as

a decomposition of the number operator process (or Poisson process in quantum

probability) into creation and annihilation parts, by analogy with the well-known

decomposition of Gaussian white noise. With help of the operator

At =

∫M

(1x6=0∇−x,t +

1

21x=0∇−0,t∇−0,t

)dµ(x), t ∈ IR+,

we associate a notion of generator and a class of partial differential equations that

can have negative second order coefficient to processes that anticipate the Levy

filtration, or are not Markovian. The “Wiener part” 12∇−0,t∇−0,t of At is identical,

after integration with respect to dt, to the Gross Laplacian on Wiener space, cf. [9],

[10]. The Ito formula is written for non-Markovian processes and we obtain the chaos

expression

∇t f(Xht+) = f ′(Xh

t )∂tB(t) +

∫M

(f(Xht + h(x, t))− f(Xh

t ))∂tN(dx, t), t ∈ IR+,

of the martingale term in the Ito formula, where (Xht )t∈IR+ is the uncompensated

process Xht =

∫M

∫ t0h(x, s)X(dx, ds). (See Prop. 9 for a precise version of this state-

ment). As for many results in anticipating stochastic calculus, the extensions are

obtained provided some regularity assumptions are made on the stochastic processes.

Another goal of this paper is to construct an anticipating stochastic calculus

for Levy processes. As in the standard Poisson case, cf. [6], [19], we obtain for the

jump part of the Levy process X two different notions of gradient and Skorohod

integral, depending on the type (space or time) of perturbation chosen.

The organization of this paper is as follows. Sect. 2 contains preliminaries

2

on Fock space and Levy processes. We recall the construction of infinitely divisible

random variables as operators on Fock space and their application to the represen-

tation of Levy processes as operator processes. This construction can be found in

the work of quantum probabilists, cf. [2], [3], [17]. Our proof uses the Ito formula

for multiple Poisson-Wiener integrals instead of the quantum probabilistic argument

which is based on the Weyl representation and commutation relations. In Sect. 3

the operators ∇, ah and their adjoints are defined. Sect. 4 presents the differ-

ent ways to perturb the Levy process by perturbation of space or time, and gives

the interpretation of these perturbations with ∇−, ∇ in terms of Fock space. In

Sect. 5 we state the extension of the Ito formula and give the chaos expansion of

its “martingale term”. In the general form of this result, some smoothness must be

imposed on the considered functionals, i.e. the formula may hold only in distribution

sense. In Sect. 6 we study the connection between our form of the Ito formula and

its associated integro-differential equations, with different examples. We also discuss

possible directions for the extension of our construction. In Sect. 7 two different Sko-

rohod integrals are constructed depending on the type of perturbation chosen (times

changes or shifts of trajectories). Their properties as extensions of the stochastic

integral are stated given a natural definition of adaptedness, unifying the different

notions of gradient and Skorohod integral on the Wiener and Poisson spaces, cf. [6],

[8], [16]. Sect. 8 deals with the Clark formula for Levy processes.

2 Representation of Levy processes as operator

processes on Fock space

This section consists essentially in definitions. Its main result is Prop. 1 which gives

the action on the Fock space of the multiplication operator by a single stochastic

integral.

2.1 Creation and annihilation on Fock space

The Fock space Γ(H) on a Hilbert space H is defined as the direct sum

Γ(H) =⊕n≥0

Hn,

where the symmetric tensor product Hn is endowed with the norm

‖ · ‖2Hn= n! ‖ · ‖2

H⊗n , n ∈ IN.

3

Here, “⊗” denotes the completed tensor product and “” denotes its symmetrization.

The annihilation and creation operators ∇− : Γ(H) −→ Γ(H)⊗H and ∇+ : Γ(H)⊗H −→ Γ(H) are densely defined as

∇−hn = nhn−1 ⊗ h, ∇+ (hn ⊗ g) = hn g, n ∈ IN, (1)

by linearity and polarization. They are closable, of domainsDom(∇−) andDom(∇+).

Let E denote the dense set of elements of Γ(H) that have a finite chaotic develop-

ment. Given h ∈ H, the closable operators a−h , a+h of quantum probability, cf. [13],

[17], are densely defined as

a−hF = (∇−F, h)H, a+hF = ∇+(F ⊗ h), F ∈ E . (2)

If A is an operator on H, the differential second quantization of A is the linear

operator dΓ(A) defined by

dΓ(A) (h1 · · · hn) =k=n∑k=1

h1 · · · Ahi · · · hn,

h1, . . . , hn ∈ Dom(A), i.e. dΓ(A)F = ∇+ (A∇−F ), and dΓ(Id) is the number opera-

tor.

2.2 Levy processes and stochastic integrals

Let M be a metric space with Borel σ-algebraM. We only assume that M contains

an element denoted by 0, with | x |= d(x, 0), so that M can be a manifold, and let

M∗ = M \ 0. Let µ be a σ-finite Radon measure on (M,M) such that∫M

|x |2∧1µ(dx) <∞,

with µ(0) = 1. Consider a Levy process of the form

X(dx, dt) = X(dx, dt)− µ(dx)dt = dBt +N(dx, dt)− µ(dx)dt,

where N(dx, dt) = N(dx, dt) − µ(dx)dt is a compensated Poisson random measure

on M∗ × IR+ of intensity µ(dx)dt, and (Bt)t∈IR+ is a standard Brownian motion

independent of N(dx, ds). The underlying probability space is denoted by (Ω,F , P ),

where F is generated by X. As a convention we set L2(M) = L2(M,µ), L2(M ×IR+) = L2(M × IR+, µ(dx)dt), and L2(Ω) = L2(Ω, P ). We define the filtration

(Ft)t∈IR+ as Ft = σ(X(dx, ds) : x ∈M, s ≤ t). A process u ∈ L2(Ω)⊗L2(M × IR+)

4

is said to be Ft-adapted if (∫Mh(x)u(x, t)dµ(x))t∈IR+ is adapted for any h ∈ Cc(M).

The integral of a square-integrable (Ft)t∈IR+-adapted process u ∈ L2(Ω)⊗L2(M×IR+)

with respect to X(dx, dt) is written as∫M×IR+

u(x, t)X(dx, dt),

with the isometry

E

[(∫M×IR+

u(x, t)X(dx, dt)

)2]

= E

[∫M×IR+

u2(x, t)µ(dx)dt

], (3)

and the multiple stochastic integral In(hn) of hn ∈ L2(M × IR+)n can be defined by

induction with

In(hn) = n

∫M×IR+

In−1(πnx,thn)X(dx, dt),

where

πnx,t : L2(M × IR+)n −→ L2(M × IR+)(n−1) (4)

is defined by[πnx,thn

](x1, t1, . . . , xn−1, tn−1) = hn(x1, t1, . . . , xn−1, tn−1, x, t)1[0,t](t1) · · · 1[0,t](tn−1),

for x1, . . . , xn−1, x ∈M and t1, . . . , tn−1, t ∈ IR+. The isometry property

E[In(hn)2

]= n! ‖ hn ‖2

L2(M×IR+)⊗n ,

follows from (3). Let h ∈ L2(M × IR+). The characteristic function of

I1(h) =

∫M×IR+

h(x, t)X(dx, dt) =

∫M∗×IR+

h(x, t)N(dx, dt) +

∫ ∞0

h(0, t)dBt

is given by the Levy-Khintchine formula

E[eizI1(h)

]= exp

(−1

2z2

∫ ∞0

h(0, t)2dt+

∫M∗×IR+

(eizh(x,t) − 1− izh(x, t))µ(dx)dt

).

2.3 Chaotic calculus

In the remaining of this paper we work on the Fock space Γ(H), with H = L2(M ×IR+), and let K = L2(M). Let C = C1

c (IR+, Cc(M)) ∩ Cc(M × IR+).

Definition 1 Let S denote the vector subspace of Γ(H) generated by elements of the

form h1 · · · hn, where h1, . . . , hn ∈ C, n ∈ IN.

5

Elements of Γ(H) are identified with random variables in L2(Ω), by associating hn ∈L2(M × IR+)n to its multiple stochastic integral In(hn), building the classical linear

isometric isomorphism from Γ(H) onto L2(Ω). For F ∈ S, ∇F ∈ Γ(H)⊗L2(M×IR+)

is identified to a square-integrable function on M × IR+ with values in S, and this

function will be denoted as (∇−x,tF )(x,t)∈M×IR+ . In Sect. 4, elements of S will be

interpreted via the Fock space isomorphism as smooth random variables that will be

defined everywhere, that is for every trajectory of the Levy process X.

We denote by π : L2(M × IR+) → L2(M∗ × IR+) the canonical projection. Let

h ∈ L2(M × IR+), let H denote the multiplication operator by the function h, and

let h = πh. We define the operator Y h on the dense domain S in Γ(H) as:

Y hF = ∇+(Hπ∇−F

)+∇+(F ⊗ h) + (∇−F, h)H.

The following is an adaptation of a result of [17] with a different proof.

Proposition 1 The operator Y h = dΓ(Hπ)+a−h +a+h on Γ (H) acts by multiplication

by I1(h) on L2(Ω) under the identification between Γ(H) and L2(Ω).

Proof. We use the Ito formula for multiple Wiener-Poisson n-th and first order

stochastic integrals, cf. e.g. [23], [24]. Let g ∈ C. We have

I1(h)In(gn) = nIn(g(n−1) (hg)) + In+1(gn h) + n(g, h)HIn−1(g(n−1)), (5)

n ≥ 1, and this identity can be rewritten as

I1(h)In(gn) = ∇+(Hπ∇−In(gn)) +∇+(In(gn)⊗ h) + (∇−In(gn), h)H. 2

This proposition will be used in Sect. 7 for the construction of the Skorohod integral.

From (5) we have S ⊂ ∩p≥2Lp(Ω). If h ∈ L∞(M × IR+) has finite measure support,

Prop. 1 gives(Φ, exp(−izY h)Φ

)Γ(L2(H))

= exp

(−1

2z2

∫ ∞0

h2(0, s)ds+

∫M∗×IR+

(eizh(x,s) − 1− izh(x, s))µ(dx)ds

), z ∈ IR,

where Φ = 1 denotes the vacuum vector in Γ(H). In other terms, the spectral

measure associated to Φ of Y h is the law of I1(h) =∫M×IR+

h(x, t)X(dx, dt).

6

3 Operators on Fock space defined by derivation

of kernels

Let 1 denotes the unit function in K. Let ∂ denote the operator of differentiation

with respect to t ∈ IR+ of differentiable functions f ∈ L2(IR+,K) and let ∂∗ be

defined on L2(IR+,K) as

∂∗u(t) =

∫ t

0

u(s)ds ∈ K, t ∈ IR+, u ∈ L2(IR+,K).

The operators ∂, ∂∗ are adjoint in the following sense:

(∂u, v)L2(IR+,K) = (u, ∂∗v)L2(IR+,K), u ∈ C, v ∈ L2(IR+,K).

Let ξ[t denote the projection in L2(IR+,K) defined as ξ[tf = 1[t,∞[f , t ∈ IR+.

3.1 Operators ∇ and ∇⊕

As a convention, tensor products are completed only if vector spaces are closed.

Definition 2 We define respectively on S and S ⊗L2(IR+) the following unbounded

operators by linearity and polarization.

• Let ∇ : Γ(L2(IR+,K)) −→ Γ(L2(IR+,K))⊗ L2(IR+) be defined by

∇t f n = −nf (n−1) (ξ[t∂f

)∈ L2(IR+,K)n, t ∈ IR+, n ∈ IN. (6)

• Let ∇⊕ : Γ(L2(IR+,K))⊗ L2(IR+,K) −→ Γ(L2(IR+,K)) be defined by

∇⊕ (f n ⊗ g) = nf (n−1) (∂ (f∂∗g)) , n ∈ IN. (7)

An operator similar to ∇ has been defined in different contexts in [11], [21]. Mim-

icking the quantum probabilistic definition (2) of the operators a−g and a+g , let ag ,

a⊕g , g ∈ L2(IR+) be defined as

ag F = (∇F, g)L2(IR+), a⊕g F = ∇⊕(F ⊗ g), F ∈ S. (8)

The definitions of ag and a⊕g as operators on S can also be extended to g ∈L2(IR+,K):

Definition 3 For g ∈ L2(IR+,K) we define on S

ag F = −∇+(∂∗g∂ ∇−F ), a⊕g F = ∇+(∂(∂∗g∇−F )) = ∇⊕(F ⊗ g). (9)

7

This definition is consistent with (8) since for g ∈ L2(IR+), a⊕g F = ∇⊕(F ⊗ g), and

ag fn = (∇f n, g)L2(IR+), as follows from the equality

ag fn = −nf (n−1) (∂f∂∗g) = −n

∫ ∞0

f (n−1) [ξ[t∂f

]g(t)dt = (∇f n, g)L2(IR+),

(10)

and ag is adjoint of a⊕g :

< ag hn, f n >Γ(H) = n < h(n−1) ∂(h∂∗g), f n >Γ(H)

= n(h, f)L2(IR+,K)(∂(h∂∗g), f)L2(IR+,K)

= −n(h, f)L2(IR+,K)(h∂∗g, ∂f)L2(IR+,K) =< hn, ag f

n >Γ(H),

f, g, h ∈ C. Consequently, ∇ and ∇⊕ are also adjoint of each other:

< ∇⊕(f n ⊗ g), h(n+1) >Γ(H)=< f n ⊗ g,∇hn >Γ(H)⊗L2(IR+),

f, g, h ∈ C. and since S is dense in L2(IR+,K), and ∇, ∇⊕, ag , a⊕g are closable.

3.2 Relationship to the number operator

We notice here that the operators ∇, ∇⊕, and the type of time perturbation they

relate to are closely connected to the number operator on Γ(L2(IR+,K)), or more

precisely to a second quantization operator. This property will be useful in Sect. 7

to distinguish between the two different notions of Skorohod integrals.

Proposition 2 For g ∈ L2(IR+,K), we have

ag + a⊕g = dΓ(g), (11)

on S, where g is identified to a multiplication operator in L2(IR+,K).

Proof. This relation is a consequence of the identity ∂(f∂∗g) − ∂∗g∂f = fg, for

f ∈ C, g ∈ L2(IR+,K), and of the definition (9) of ag and a⊕g .2

This decomposition can be viewed as a decomposition of Poisson noise into creation

and annihilation parts, in a way that parallels the well-known decomposition of white

noise.

8

3.3 Product rule

In this subsection elements of Γ(H) are identified with random variables in L2(Ω) via

the Wiener-Poisson-Ito isometric isomorphism, hence (∇−x,tF )(x,t)∈M×IR+ is an element

of L2(M × IR+, L2(Ω)), for F ∈ Dom(∇−).

Remark 1 Due to the Ito formula (5), every element of S can be expressed as a

polynomial in single stochastic integrals with respect to X. Conversely, a polynomial

in single stochastic integrals with respect to X is in S, provided its integrands are in

C.

As will be recalled in the next section, ∇−0,t is identified to a derivation operator,

and ∇−x,t is, for x 6= 0, is identified to a finite difference operator. Hence we have the

identity

∇−x,t(FG) = F∇−x,tG+G∇−x,tF + 1x 6=0∇−x,tF∇−x,tG, (x, t) ∈M∗ × IR+, (12)

F,G ∈ S.

Proposition 3 The action of the operator ag on a product is the following:

ag (FG) = Fag G+Gag F −∫M×IR+

g(x, t)∇−x,tF∇−x,tGµ(dx)dt, (13)

g ∈ L2(M × IR+), F,G ∈ S.

Proof. The proof of this result uses the Ito formula, or equivalently Prop. 1. Let

h, f ∈ C, and let h denote the function h = πh, where π : L2(M × IR+)→ L2(M∗ ×IR+) is the canonical projection. We have

ag (I1(h)In(f n)) +

∫M×IR+

g(x, t)∇−x,tI1(h)∇−x,tIn(f n)µ(dx)dt

= −nIn(f (n−1)

[(∂fh+ f∂h)∂∗g

])−n(n− 1)In

(f (n−2) hf ∂f∂∗g

)− nIn+1

(f (n−1) h ∂f∂∗g

)−In+1 (f n ∂f∂∗g)− n(n− 1)(f, h)HIn−1

(f (n−2) ∂f∂∗g

)+n

∫M×IR+

g(x, t)h(x, t)f(x, t)µ(dx)dtIn−1(f (n−1))

= −n(n− 1)In

(f (n−1) fh ∂f∂∗g

)− nIn

(f (n−1) h∂f∂∗g

)−nIn+1

(f (n−1) h ∂f∂∗g

)− n(n− 1)(f, h)HIn−1

(f (n−2) ∂f∂∗g

)−n(∂∗g, h∂f)HIn−1

(f (n−1)

)−nIn

(f (n−1) ∂h∂∗gf

)− In+1 (f n ∂h∂∗g)− n(∂∗g, f∂h)HIn−1(f (n−1))

= I1(h)ag In(f n) + In(f n)ag I1(h).

9

The formula is then easily extended by induction to F,G ∈ S from Prop. 1, as in

Prop. 8 of [22]. Assume that (13) holds for F = In(f n) and G = I1(h)k for some

k ≥ 1. Then using the identity (12) satisfied by ∇−x,t, we can write

ag (In(f n)I1(h)k+1)

= I1(h)ag (In(f n)I1(h)k) + In(f n)I1(h)kag I1(h)

−∫M×IR+

g(x, t)∇−x,tI1(h)∇−x,t(I1(h)kIn(f n))µ(dx)dt

= I1(h)(I1(h)kag In(f n) + In(f n)ag

(I1(h)k

))−I1(h)

∫M×IR+

g(x, t)∇−x,t(I1(h)k

)∇−x,tIn(f n)µ(dx)dt+ In(f n)I1(h)kag I1(h)

−∫M×IR+

g(x, t)∇−x,tI1(h)(I1(h)k∇−x,tIn(f n) + In(f n)∇−x,t

(I1(h)k

))µ(dx)dt

−∫M×IR+

1x 6=0g(x, t)∇−x,tI1(h)∇−x,tI1(h)k∇−x,tIn(f n)µ(dx)dt

= I1(h)k+1ag In(f n) + In(f n)ag(I1(h)k+1

)−

∫M×IR+

g(x, t)∇−x,t(I1(h)k+1

)∇−x,tIn(f n)µ(dx)dt.

2

Following the proof of Prop. 3 we can show

∇t (FG) = F∇t G+G∇t F −∫M

∇−x,tF∇−x,tGµ(dx), F,G ∈ S, t ∈ IR+, (14)

hence for g ∈ L2(Ω)⊗ L2(IR+),

ag (FG) = Fag G+Gag F −∫ ∞

0

g(t)

∫M

∇−x,tF∇−x,tGµ(dx)dt, (15)

a.s., F,G ∈ S. However this formula is not extended to a random g ∈ L2(Ω) ⊗L2(M × IR+) since ag is not defined for such processes. See Sect. 7 for an extension

of the definition to random g.

4 Perturbations of Levy processes and their Fock

space interpretation

In this section we study the probabilistic interpretations of ∇− and ∇. While it is

well known that ∇− is interpreted by shifts of trajectories on both the Wiener and

Poisson spaces, we show that ∇ corresponds to perturbations by time changes.

10

4.1 Perturbations by shifts of trajectories

First by perturbation via addition of a jump to the Poisson point measure and

infinitesimal shift of the Brownian trajectory, we get the annihilation operator on

Γ(H).

Proposition 4 We have for F ∈ S:

∇−x,tF = F (X(·) + δx,t(·))− F, (x, t) ∈M∗ × IR+, (16)

and

∇−0,tf (I1(h1), . . . , I1(hn)) =i=n∑i=1

hi(0, t)∂if (I1(h1), . . . , I1(hn)) , (17)

f ∈ C1b (IR

n), h1, . . . , hn ∈ C, t ∈ IR+, or more formally:

∇−0,tF = limε→0

1

ε(F (X(·) + εδ0,t(·))− F ) , F ∈ S.

Proof. The Wiener and Poisson parts of X can be treated separately. In the Wiener

case we refer to [14], [27] and to the references therein. In the Poisson case this result

is contained in [8], [16].2

The Wiener part (∇0,tF )t∈IR+ of the operator ∇− is also called the Malliavin deriva-

tive, cf. [14]. The following relation between the values of the gradients ∇− on

M∗× IR+ and on 0× IR+ is connected to the convergence of the renormalized Pois-

son process to Brownian motion, and shows that ∇−F on 0× IR+ can be obtained

by continuity from its values on M∗ × IR+.

Proposition 5 Let h ∈ Cc(M) with limx→0 h(x) = 0 and h(x) 6= 0, ∀x ∈ M∗. If

F ∈ Dom(∇−) is of the form F = f(∫

M

∫ t0h(x)X(dx, ds)

), f ∈ C1

b (IR), h ∈ Cc(M),

then

∇−0,sF = limx→0

1

h(x)∇−x,sF, s ∈ IR+, a.s.

Proof. This is a consequence of Relations (16) and (17).2

11

4.2 Perturbations by time changes

We now turn to the probabilistic interpretation of ∇ and ∇⊕. The main difference

between our construction and that of e.g. [1] is that only jump times are perturbed

in the Poisson part of X. In this way we retain the connection between variational

calculus and stochastic integration, cf. Sect. 7. Since this probabilistic interpretation

will involve time changes and such perturbations can not be absolutely continuous

with respect to the Wiener measure (they can be, however, with respect to the Pois-

son measure), we will consider functionals that can be defined everywhere, i.e. for

every trajectory of X. Single stochastic integrals with respect to a Poisson measure

can be defined everywhere provided the support of the integrand has finite intensity

measure. Wiener single stochastic integrals can be defined for every Brownian trajec-

tory if the integrand is continuously differentiable in the time variable so as to allow

to write an integration by parts formula. Hence single stochastic integrals in S can

be defined trajectory by trajectory, but S also contains iterated stochastic integrals

whose definition trajectory by trajectory is a priori ambiguous. We choose to define

them everywhere by taking into account Remark 1 and by using the expression of

elements of S as polynomials in single stochastic integrals. More precisely, we state

the following definitions.

Definition 4 Let F be a random variable defined for every trajectory of X. For

h ∈ L2(M × IR+) ∩ L∞(M × IR+) with ‖ h ‖L∞(M×IR+)< 1, let ThF denote the

functional F ∈ S evaluated at time-changed trajectories whose jumps are obtained

from the jumps of N(dx, ds) via the mapping

M × IR+ −→M × IR+

(x, t) 7→ (x, νh(x, t)) = (x, t+ ∂∗h(x, t)),

and whose continuous part is given by the time-changed Brownian motion (Bht )t∈IR+

defined as

Bhνh(0,t) = Bt, t ∈ IR+.

Since most functionals of stochastic analysis are only defined almost surely, we will

also need the following.

Definition 5 Let D denote the vector space dense in L2(Ω) generated by

In(h1 · · · hn) : h1, . . . , hn ∈ ∩p≥2Lp(M × IR+), n ∈ IN.

12

Let h ∈ L2(M × IR+) ∩ L∞(M × IR+) with ‖ h ‖L∞(M×IR+)< 1. For F ∈ D of the

form In(f1 · · · fn), let F = f(I1(g1), . . . , I1(gm)) denote the expression of F as a

polynomial in single stochastic integrals obtained from Prop. 1. We define

UhF = f(I1(g1 νh), . . . , I1(gm νh)).

The definition of Uh extends to D by linearity.

The interest in the operator Uh, compared to Th, is that it is defined on a set of L2

functionals, whereas Th is not. The link between Uh and Th is given by the following

remark.

Remark 2 For any F ∈ S there is a version F of F such that UhF = ThF , a.s.

Proof. It suffices to do the proof in the Wiener case, for F ∈ S of the form F = I1(f).

It this case, I1(f) =∫∞

0f ′(s)Bsds, a.s., hence letting F =

∫∞0f ′(s)Bsds, we obtain

ThF =

∫ ∞0

f ′(s)B(ν−1h (s))ds =

∫ ∞0

f ′(νh(s))Bs(1+h(s))ds =

∫ ∞0

f(νh(s))dBs, a.s.,

hence ThF = UhF , a.s.2

Proposition 6 Let u ∈ L2(M × IR+) ∩ L∞(M × IR+). For F ∈ S we have

− d

dεUεuF|ε=0 = auF +

∫M×IR+

u(x, t)

(1x6=0∇−x,t +

1

21x=0∇−0,t∇−0,t

)Fµ(dx)dt,

(18)

the limit being taken in L2(Ω).

Proof. Relation (18) is proved in two steps. First we notice that it holds for a simple

stochastic integral I1(h), h ∈ C, cf. Prop. 9 of [22] and [21], and then use the product

rules (13) and (15) which imply that

F 7→ auF +

∫M×IR+

u(x, t)

(1x 6=0∇−x,t +

1

21x=0∇−0,t∇−0,t

)Fµ(dx)dt

is a derivation operator on S, given that ∇−0,t is a derivation operator and that ∇−x,tfor x 6= 0 satisfies as a finite difference operator:

∇−x,t(FG) = F∇−x,tG+G∇−x,tF +∇−x,tF∇−x,tG, (x, t) ∈M∗ × IR+.

13

2

Prop. 6 will be interpreted in Sect. 5 as an extended form of the Ito formula, in which

ag F , having expectation zero, represents a “martingale term”. Although it is dense

in L2(Ω), the class S is too small to be of real interest in stochastic analysis since it

does not contain the increments of X. Thus we need to extend Prop. 6 to a wider

class of functionals.

Definition 6 We define the operator A : D → L2(Ω)⊗ L2(IR+) by

AsF =

∫M

(1x 6=0∇−x,sF +

1

21x=0∇−0,s∇−0,sF

)µ(dx), dP ⊗ ds a.e.

The operator A, (whose “Wiener part” is the Gross Laplacian after integration with

respect to ds), will be used to define a notion of pseudo generator for non-Markovian

processes that will, due to Relation (24) below, extend the classical notion of gen-

erator. The operator A is “intrinsic”, in that unlike classical generators, it is not

determined by a particular process. Absolutely continuous drifts are not considered

here because their influence is of a deterministic nature and for this reason they do

not create new problems in an extension of stochastic calculus to an anticipative or

to a non-Markovian setting. Similarly, in order to simplify the exposition, stochas-

tic integrals with respect to the Poisson measure are evaluated for functions with

finite intensity measure support. The general case can be treated by introduction of

appropriate compensators.

Proposition 7 We have for u ∈ C1c (IR+) and F ∈ D:

− d

dε< UεuF,G >|ε=0=< F, a⊕uG >L2(Ω) + < AF,G⊗u >L2(Ω)⊗L2(IR+), G ∈ S. (19)

Proof. By comparison with the Malliavin calculus by space perturbation of trajec-

tories, the difficulty lies here in the fact that on Wiener space the transformation

Tεu is not absolutely continuous. By polarization and use of the Ito formula (5) it is

sufficient to prove (19) for F = I1(f)n, where f ∈ ∩p≥2Lp(M × IR+). We have the

chaos expansion

I1(f)n =

j=n∑j=0

Ij(fj).

Due to the multiplication formula for multiple Wiener-Poisson stochastic integrals,

fk is of the form fk = hk1 · · · hkk×Pk(f), where hk1, . . . , hkk are powers of f of degree

14

lower than 2n, and Pk(f) is a polynomial in integrals on M× IR+ of powers of f . For

h ∈ Lp(M×IR+), p ≥ 2, we let h(νεu) be the function defined by (x, t) 7→ h(x, νεu(t)).

For ε in a certain neighborhood of zero, (x, t) 7→ (x, νεu(t)) is invertible and absolutely

continuous with bounded Radon-Nykodim derivative, hence h(νεu) is well-defined in

Lp(M × IR+). We have

UεhF =

j=n∑j=0

Pj(f(νεu))Ij(hj1(νεu) · · · hjn(νεu)).

We assume that G is in the k-th chaos, k ≤ n, and that it is written as G = Ik(gk),

g ∈ C. With this notation,

< UεuF,G >L2(Ω)= Pk(f(νεu))(hk1(νεu) · · · hkk(νεu), gk)L2(M×IR+)⊗k . (20)

Now,

(hk1(νεu) · · · hkk(νεu), gk)L2(M×IR+)⊗k

=

(hk1,

(g

1 + εu

)(ν−1εu )

)L2(M×IR+)

· · ·(hkk,

(g

1 + εu

)(ν−1εu )

)L2(M×IR+)

, (21)

and the derivative of(

g1+εu

)(ν−1εu ) in ε is continuously differentiable and uniformly

bounded on M × IR+ for ε in a neighborhood of zero by a function integrable on

M × IR+. An analogous change of variables can be performed in Pk(f), hence <

UεuF,G > is differentiable in ε in a certain neighborhood of zero. For F ∈ S,

Relation (19) is a consequence of Prop. 6 and of the duality between ∇ and ∇⊕.

In order to prove (19) for F ∈ D we need to exchange the derivation with respect

to ε with the limit of a sequence (Fn)n∈IN in S such that (Fn)n∈IN and (AFn)n∈IN

converge respectively in L2(Ω) and L2(Ω) ⊗ L2(IR+) to F ∈ D and to AF . Hence

the proposition will hold if we show

d

dε< UεuF,G >|ε=0= lim

n→∞

d

dε< UεuFn, G >|ε=0, G ∈ S.

For this it is sufficient to prove the following, given the polynomials expressions (20),

(21) . Let N ≥ 1 and let (αn)n∈IN be a sequence in C converging in ∩p=Np=2 Lp(M × IR)

to α. We have for ε in a neighborhood of zero

d

dε(α(νε), g)L2(M×IR+)

=d

dε(α,

(g

1 + εu

)(ν−1εu ))L2(M×IR+) = (α,

d

dε

(g

1 + εu

)(ν−1εu ))L2(M×IR+)

15

= limn→∞

(αn,d

dε

(g

1 + εu

)(ν−1εu ))L2(M×IR+) = lim

n→∞

d

dε(αn,

(g

1 + εu

)(ν−1εu ))L2(M×IR+)

= limn→∞

d

dε(αn(νεu), g)L2(M×IR+).

2

Prop. 7 also gives:

− d

dε< Uεuf(I1(h)), G >|ε=0 = < f(I1(h)),∇⊕(G⊗ u) >L2(Ω)

+ <

∫ ∞0

us[Ghs f

](I1(h))ds,G >L2(Ω),

G ∈ S, for f polynomial, h ∈ ∩p≥2Lp(M × IR+), u ∈ C1

c (IR+). For G = 1 we can

along the lines of the proof of Prop. 7 show the following result.

Proposition 8 Let T ∈ IR+ and let (uε)ε∈IR+ ⊂ C1c ([0, T ]) be continuous in ε for the

‖ · ‖L∞(IR+) norm. We have for F ∈ D:

d

dεE[UεuεF ]|ε=0 = E[(AF, u0)L2(IR+)].

5 A chaos approach to the Ito formula

The aim of this section is to develop from Prop. 6 a formula that extends the Ito

formula and gives the chaos expansion of its martingale term. The generator (Ghs )s∈IR+

of the uncompensated process

Xht =

∫M∗

∫ t

0

h(x, s)N(dx, ds) +

∫ t

0

h(0, s)dBs, t ∈ IR+,

where h ∈ L2(M × IR+) has finite measure support, is given by[Ghs f

](x) =

∫M∗

(f(x+ h(y, s))− f(x))µ(dy) +1

2h(0, s)2∂2f(x), x, s ∈ IR+, (22)

f ∈ C2(IR). The Dynkin formula says that the process f(Xht )− f(0)−

∫ t0Ghs f(Xh

s )ds

is a martingale relative to the filtration (Ft)t∈IR+ , and the Ito formula identifies this

martingale:

f(Xht )− f(0)−

∫ t

0

[Ghs f

](Xh

s )ds

=

∫ t

0

h(0, s)f ′(Xhs )dBs +

∫M∗

∫ t

0

(f(Xhs + h(x, s))− f(Xh

s ))N(dx, ds), (23)

16

f ∈ C2(IR). Note that from (16), (17) and (22) we have the relations

[Ghs f

](I1(h)) =

∫M

[1x 6=0∇−x,s +

1

21x=0∇−0,s∇−0,s

]f(I1(h))µ(dx), (24)

and

1[0,t](s)(Ghs f

)(Xh

t ) =

∫M

[1x 6=0∇−x,s +

1

21x=0∇−0,s∇−0,s

]f(Xh

t )µ(dx), (25)

for f ∈ C2b (IR). A similar relation can be written if Xh is replaced by a diffusion

process but it is not as straightforward, cf. Relation (29) in Sect. 6. The annihilation

operator∇− does not appear only in the generator Ghs , but also in the martingale term

of the Ito formula from its expressions (16) and (17), moreover the anticipating Ito

formula makes use of ∇− in the Wiener case, cf. [14], [26]. However, the martingale

term (23) can not be explicitly written with ∇−. The closest result that directly

uses ∇− may be the Clark formula, cf. [7], [25] and Sect. 8 for its extension to Levy

processes. The aim of the following lemma is to provide a chaos form for the Ito

formula, using the operators ∇ and ∇−.

Lemma 1 Let h ∈ C, and let u ∈ L2(Ω× IR+) ∩ L∞(Ω× IR+) be (Ft)-adapted. We

have for f polynomial:

− d

dεUεuf(I1(h))|ε=0 = (∇f(I1(h)), u)L2(IR+) + (Ghf(I1(h)), u)L2(IR+). (26)

Proof. We apply Prop. 6 and use Relation (24).2

Each term in the above Lemma belongs to L2, but the smoothness imposed on

functionals and the type of perturbation chosen do not make obvious the analogy

with the Ito formula. The following proposition gives from Lemma 1 more precise

information on the links between Ito differentials and the chaotic calculus induced

by ∇. The formula applies to Xht which does not have the smoothness property

required in Lemma 1, without having recourse to generalized random variables.

Proposition 9 Let h ∈ C, and Xht =

∫M

∫ t0h(x, s)X(dx, ds), t ∈ IR+. We have for

f polynomial

limn→∞

∫ t

0

∇s f(Xh,ns )ds =

∫ t

0

f ′(Xhs )dBs+

∫ t

0

∫M

(f(Xhs +h(x, s))−f(Xh

s ))N(dx, ds),

17

where the limit is taken in L2(Ω), and (Xh,n)n∈IN is any sequence converging to Xh

in L2(Ω)⊗ L2(IR+), of the form

Xh,nt =

∫M

∫ ∞0

h(x, s)en(s− t)X(dx, ds), n ∈ IN, t ∈ IR+,

where (en)n∈IN ⊂ C1b (IR) converges pointwise to 1]−∞,0] with en(s) = 1, s ≤ 0, 0 ≤

en(s) ≤ 1, s ≥ 0, n ∈ IN.

Proof. Let (en)n∈IN be a sequence of smooth positive functions bounded by one and

converging everywhere to 1]−∞,0], with e′n = 0 on IR−. We have almost surely

− ∂

∂sXh,ns =

∫M

∫ ∞0

h(x, u)e′n(u− s)X(dx, du) =

∫M

∫ ∞s

h(x, u)e′n(u− s)X(dx, du)

=

∫M

∫ ∞s

∂

∂u(h(x, u)en(u− s))(u)X(dx, du)−

∫M

∫ ∞s

∂h(x, u)en(u− s)X(dx, du),

which implies

∂

∂sf(Xh,n

s ) = −∫M

∫ ∞s

∂

∂u(h(x, u)en(u− s))X(dx, du)f ′(Xh,n

s )

+

∫M

∫ ∞s

∂h(x, u)en(u− s)X(dx, du)f ′(Xh,ns ).

With the same argument as in Prop. 6 we can show, using (14) and (24), that

−∫M

∫ ∞s

∂

∂u(h(x, u)en(u− s))X(dx, du)f ′(Xh,n

s ) = ∇s f(Xh,ns ) + Ghs f(Xh,n

s ),

hence

∂

∂sf(Xh,n

s ) = ∇s f(Xh,ns ) + Ghs f(Xh,n

s ) +

∫M

∫ ∞s

∂h(x, u)en(u− s)X(dx, du)f ′(Xh,ns ),

and by integration on [0, t]:

f(Xh,nt ) = f(0) +

∫ t

0

∇s f(Xh,ns ) +

∫ t

0

Ghs f(Xh,ns )ds

+

∫ t

0

∫M

∫ ∞s

∂h(x, u)en(u− s)X(dx, du)f ′(Xh,ns )ds.

It remains to take the limit in L2(Ω), which does not depend on the choice of the

sequence (en)n∈IN as n goes to infinity, and to use (23).2

The result of the above proposition might be formally written as

∇t f(Xht+) = f ′(Xh

t )∂tB(t) +

∫M

(f(Xht + h(x, t))− f(Xh

t ))∂tN(dx, t),

18

where ∂tB(t), ∂tN(dx, t) denote the Gaussian and Poissonian white noises.

We will now write a statement which is closer than Lemma 1 to the classical

Ito formula, and applies to a class of processes that can be anticipating with respect

to the Levy filtration, or non-Markovian. More generally, the considered processes

do not need to possess any particular property with respect to a filtration. We state

below the properties that should be satisfied by (Xt)t∈IR+ in order to extend the Ito

formula and the notion of generator to this process. Apart from the smoothness and

integrability hypothesis (1) and (2), the third condition ensures a form of consistency

in the time evolution of the process without requiring it to be Markov or adapted.

Definition 7 We denote by V the class of processes (Xt, (utε)ε∈IR+)t∈IR+ where (Xt)t∈IR+

is a family of random variables and (utε)t,ε∈IR+ is a family of functions such that for

any t > 0,

1. Xt ∈ D,

2. (utε)ε∈IR+ is continuous in ε for the ‖ · ‖L∞(IR+) norm,

3. for some T t ∈ IR+, utε ∈ C1c ([0, T

t]) and satisfies UεutεXt = Xt−ε a.s., for ε in a

neighborhood of zero.

The family (utε)t,ε∈IR+ may be independent of ε, and in this case we use the notation

(Xt, ut)t∈IR+ ∈ V .

Theorem 1 Let (Xt, ut)t∈IR+ ∈ V be such that Xt ∈ S, ∀t ∈ IR+. We have the

extension of the Ito formula

f(Xt) = f(0) +

∫ t

0

(∇f(Xs), us)L2(IR+)ds+

∫ t

0

(Af(Xs), us)L2(IR+)ds, (27)

for f polynomial.

Proof. This relation is (in differential form) a consequence of Def. 7 and Prop. 6.2

In (27) the “martingale” term∫ t

0(∇f(Xs), u

s)L2(IR+)ds is actually a finite variation

process since (Xt)t∈IR+ ⊂ S. In the general case, this process is obviously not a

martingale but it has expectation zero and by analogy with classical diffusions, a

“martingale property” could be written here as

E

[∫ t

0

(∇f(Xs), us)L2(IR+)ds | Fv

]= E

[∫ t

0

(1[0,v]∇f(Xs), us)L2(IR+)ds | Fv

],

v ∈ IR+, given the property E[∇s F | Ft] = 0 of ∇, s < t, obtained from (6).

19

6 Pseudo generators of non-Markovian stochastic

processes

In this section we show that Prop. 8 yields a systematic method to find a partial

differential or integro-differential equation associated to the law of non-Markovian

stochastic processes. In the Markov case the results coincide with the ones obtained

via the classical theory. From the operator A we construct a pseudo generator Lassociated to a stochastic process in the class V of Def. 7.

Definition 8 To any process (Xt, (utε)ε∈IR+)t∈IR+ in V we associate a family (Lt)t∈IR+

of operators defined as

(Ltf)(Xt) = E[(Af(Xt), u

t0)L2(IR+) | Xt

],

for f polynomial, t ∈ IR+.

Since (Af(Xt), ut0)L2(IR+) ∈ L1(Ω), Ltf is defined dpt-a.e., where pt is the law of Xt.

The following result uses the operator L to extend the notion of generator.

Theorem 2 Let (Xt, (utε)ε∈IR+)t∈IR+ ∈ V. The law pt of Xt satisfies the integro-

differential equation∂

dt

∫IR

fdpt =

∫IR

Ltfdpt, t ∈ IR+, (28)

for f polynomial.

Proof. Relation (28) is a direct consequence of Def. 8 and Prop. 8 that give

d

dtE[f(Xt)] = E[(Af(Xt), u

t)], t ∈ IR+. 2.

Relation (28) can be written in distribution sense as ∂∂tpt(x) = L∗tpt(x), t ∈ IR+,

x ∈ IR, where L∗t denotes the adjoint of Lt. In view of the remark at the end of

Sect. 4 it is also possible to choose ut ∈ L2(M × IR+). In this way we can, for

instance in case X is a sum of independent Poisson processes, perturb each process

independently.

Until the end of this section we focus on the continuous case, in which it is

possible to gain more information on the operator Lt. We have

Asf(Xt) =1

2∇−0,s(f ′(Xt)∇−0,sXt) =

1

2f ′′(Xt)(∇−0,sXt)

2 +1

2f ′(Xt)∇−0,s∇−0,sXt.

Hence Lt is of the form

Lt = at(x)∂2x + bt(x)∂x,

20

where at and bt are functions defined pt-a.e. on IR as

at(Xt) =1

2E[((∇−0,·Xt)

2, ut)L2(IR+) | Xt], bt(Xt) =1

2E[(∇−0,·∇−0,·Xt, u

t)L2(IR+) | Xt].

We now illustrate Th. 2 with examples whose choices relies essentially on

the simplicity of calculations. In most cases, various differential equations can be

written for the law pt of a stochastic process, even when the Ito formula is not directly

applicable. The point in the present method is that it gives a systematic procedure

to derive a differential equation which is “canonical” in that its coincides with the

result obtained via the Ito formula in the Markov case. For each a, b ∈ IR+, a < b, we

choose a function ea,b ∈ C∞c (IR+) with support in [a, b] and such that∫ baea,b(x)dx = 1,

with ea,a = 0.

• The main interest in this example is that it does not require the computation

of a conditional expectation, hence it clearly shows the role played by ∇−. Let

Xt = B2t + (B2t − Bt)

2, t ∈ IR+. The process (Xt)t∈IR+ is not Markovian, but it

has same law as a squared Bessel process. Let ut = e0,t/2 + et,3t/2, t > 0. We have

UεutXt = Xt−ε, hence (Xt, ut)t∈IR+ ∈ V , and

Asf(Xt) =1

2∇−0,s∇−0,sf(Xt)

=1

2∇−0,s

(21[0,t](s)Btf

′(Xt) + 21[t,2t](s)(B2t −Bt)f′(Xt)

)= 1[0,2t](s)(f

′(Xt) + 2B2t 1[0,t](s) + 2(B2t −Bt)

21[t,2t](s))f′′(Xt),

hence (Af(Xt), ut)L2(IR+) = 2Xtf

′′(Xt)+2f ′(Xt), and Lt = 2x∂2x +2∂x. We retrieved

in this way the partial differential equation ∂tft = 2x∂2xft + 2∂xft satisfied by the

density ft of the law pt of (Xt)t∈IR+ .

• This example shows that the coefficient of the second order derivative term

of the pseudo generator is allowed to be negative in our approach. Consider the

process Xt =∫ b(t)a(t)

h(s)dBs where a, b ∈ C1(IR+), 0 ≤ a < b, and h ∈ L∞(IR+) is a

step function h =∑i=n

i=1 αi1[ti,ti+1[, and define utε as

utε =i=n∑i=1

1]ti,ti+1[(a(t))

(a(t+ ε)− a(t)

εeti,a(t) −

a(t+ ε)− a(t)

εea(t),ti+1

)+1]ti,ti+1[(b(t))

b(t+ ε)− b(t)ε

eti,b(t),

21

for ε > 0, and for ε = 0 as:

ut0 =i=n∑i=1

1]ti,ti+1[(a(t))(a′(t)eti,a(t) − a′(t)ea(t),ti+1

)+ 1]ti,ti+1[(b(t))b

′(t)eti,b(t).

With this definition of utε we have UεutεXt = Xt−ε,Asf(Xt) = 121[a(t),b(t)](s)h(s)f ′′(Xt),

and

(Af(Xt), ut0)L2(IR+) =

1

2(−a′(t)h2(a(t)) + b′(t)h2(b(t)))∂2f(Xt),

hence the pseudo generator of (Xt)t∈IR+ is

Lt =1

2(−a′(t)h2(a(t)) + b′(t)h2(b(t)))∂2.

• In the following example the computation of a conditional expectation is

needed in order to calculate Lt. Let Xt = Bt(B1−Bt), t ∈ [0, 1]. With ut = e0,t−et,1we have UεuXt = Xt−ε, and

(Af(Xt), ut)L2(IR+) =

1

2

((B1 −Bt)

2 −B2t

)f ′′(Xt).

The conditional expectation E[X2 | XY ] where X, Y are independent centered gaus-

sian random variables with variances a2, b2 respectively can be be computed as

E[X2 | XY ] =a

b| XY |

K1

(|XY |ab

)K0

(|XY |ab

) ,where Kν(x) is the modified Bessel function of the second kind and of order ν ∈ IN,

hence

Lt =1

2| x |

(√1− tt−√

t

1− t

)K1

(| x | /

√t(1− t)

)K0

(| x | /

√t(1− t)

)∂2x.

To end this section we discuss possible directions for the extension of the above re-

sults. The first problem that occurs is that the operator ∇ requires the Fock kernels

it acts on to be smooth. Hence a construction involving distributions can be useful to

generalize the theory, and the Hida calculus seems to be a natural tool here because

the operator ∇ becomes continuous in Sobolev spaces of Fock kernels as it acts by

derivation of these kernels. Since the expectation of ∇ in the Ito formula is zero,

22

the construction of pseudo-generators can be done without smoothness assumptions

kernels, for example on the space D. In this case the remaining problem is with the

proper definition of the operator

As =

∫M

(1x 6=0∇−x,s +

1

21x=0∇−0,s∇−0,s

)µ(dx),

which requires give a meaning to the contraction of a function of two variables. The

right definition may consist in taking right limits, as suggests the following formal

treatment of classical diffusions. Let (Xt)t∈IR+ be defined by the stochastic differential

equation

Xt =

∫ t

0

σ(Xs)dBs, t ∈ IR+, σ ∈ C2c (IR).

Here Xt /∈ D, and we indicate how the usual result can be formally recovered. We

have Xt ∈ Dom(∇−), cf. [14], and

∇−s f(Xt) =

(∫ t

s

∇−s σ(Xv)dBv + 1[0,t](s)σ(Xs)

)f ′(Xt),

and for u > s,

∇−u∇−s f(Xt) =

∫ t

s

∇−u∇−s σ(Xv)dBvf′(Xt)

+

(∫ t

s

∇−s σ(Xv)dBv + 1[0,t](s)σ(Xs)

)×(∫ t

u

∇−u σ(Xv)dBv + 1[0,t](u)σ(Xu)

)f ′′(Xt),

hence we can define

Af(Xt) =1

2lims→t−

limu→s+

∇−u∇−s f(Xt) =1

2σ2(Xt)f

′′(Xt+ε), (29)

and obtain L = 12σ2(x)∂2

x.

7 Construction of the Skorohod integral

The purpose of this section is to construct the two different Skorohod integrals in-

duced by the operators ∇+ and ∇⊕ as adjoints of gradient operators, depending on

the type (space or time) of perturbation chosen for the Poisson process. Given the

identification between Γ(L2(M × IR+)) ⊗ L2(M × IR+) and L2(Ω) ⊗ L2(M × IR+)

via multiple stochastic integrals, the following natural definition of adaptedness in

Γ(L2(IR+,K))⊗L2(IR+,K) coincides with the definition of adaptedness with respect

to the filtration (Ft)t∈IR+ generated by X.

23

Definition 9 A process (ut)t∈IR+ with values in Γ(L2(IR+,K)) ⊗ K is said to be

adapted if

(ut, hn)Γ(L2(IR+,K))⊗K = 0,

whenever hn ∈ L2([t,∞[,K)n ⊗K, n ≥ 0, t ∈ IR+.

The set of simple processes defined below is dense in Γ(L2(IR+,K)) ⊗ L2(IR+,K) 'L2(Ω)⊗ L2(M × IR+).

Definition 10 We say that u ∈ Γ(L2(IR+,K))⊗ L2(IR+,K) is a simple process if it

is written asi=n∑i=1

Fiui,

where u1, . . . , un ∈ Cc(M × IR+), and F1, . . . , Fn ∈ S, n ≥ 1. This set of processes is

denoted by U .

The first definition of the Skorohod integral uses the operator ∇+, and is the most

frequently used on the Wiener space, cf. [14], [15]. Let π : L2(M×IR+)→ L2(M∗×IR)

denote the canonical projection.

Proposition 10 Let u ∈ U be a simple process in L2(Ω, P ) ⊗ L2(M × IR+). We

have

∇+(u) =

∫M×IR+

u(x, t)X(dx, dt)−∫M×IR+

∇−x,tu(x, t)µ(dx)dt

−∇+(π(∇−· u(·)

)), (30)

and if u ∈ L2(Ω)⊗ L2(M × IR+) is (Ft)-adapted, then

∇+(u) =

∫M×IR+

u(x, t)X(dx, dt).

Proof. We work for a process of the form u = Fh and we use Prop. 1 to express

the multiplication of F ∈ S by I1(h) as a sum of three terms including ∇+(F ⊗ h),

and obtain (30). This relation is then extended to U by linearity. The definition of

adaptedness implies that the correction terms vanish in the adapted case.2

The second notion of Skorohod integral uses the operator ∇⊕ and the splitting of

dΓ(g) into ag and a⊕g in Prop. 2. If u ∈ U is a simple process written as u =∑i=n

i=1 Fiui

24

we define the operator Du on S by time changes and infinitesimal shifts of Brownian

motion from Prop. 4 and Prop. 6 as

DuF = (∇−F, u)L2(M×IR+) +i=n∑i=1

FiaπuiF, F ∈ S.

The operators D and δ = ∇+ +∇⊕ π are adjoints in the following sense. We have

E[DuF ] = < ∇−F, u >Γ(H)⊗H +E[i=n∑i=1

FiaπuiF ]

= < F,∇+(u) >Γ(H) +E[Fi=n∑i=1

a⊕πuiFi]

= < F,∇+(u) >Γ(H) +E[Fi=n∑i=1

∇⊕(πuiFi)]

= < F,∇+(u) +∇⊕(πu) >Γ(H)=< F, δ(u) >Γ(H), u ∈ U , F ∈ S.

For u ∈ U with u =∑i=n

i=1 Fiui, we define trace(Du) as

trace(Du) =

∫ ∞0

∇−s usds+i=n∑i=1

aπuiFi.

Proposition 11 If u ∈ U is a simple process in L2(Ω)⊗ L2(M × IR+), then

δ(u) =

∫M×IR+

u(x, t)X(dx, dt)− trace(Du).

If moreover u ∈ L2(Ω)⊗ L2(M × IR+) is (Ft)-adapted, then

δ(u) = ∇+(u) =

∫M×IR+

u(x, t)X(dx, dt).

Proof. For u of the form u = Fh we use Prop. 10 and the decomposition of the

number operator that follows from (9), (10) and (11):

∇+(π∇−· u·) = ∇+(πh·∇−· F ) = aπhF + a⊕πhF = aπhF +∇⊕(πu).

Finally we use the fact that ∇⊕ vanishes on adapted processes from its definition

(7).2

25

8 Clark formula

In this section we extend the Clark formula, cf. [7], [25], to the case of Levy processes.

Proposition 12 For F ∈ L2(Ω), we have

F = E[F ] +

∫M×IR+

E[∇−x,tF | Ft]X(dx, dt).

Proof. Let ∆n = ((x1, t1), ..., (xn, tn)) ∈ (M × IR+)n : t1 < · · · < tn. We have for

F ∈ S:

F = E[F ] +∑n≥1

In(fn1∆n)

= E[F ] +∑n≥1

∫M×IR+

In−1(fn(·, x, t)1∆n(·, x, t))X(dx, dt)

= E[F ] +

∫M×IR+

∑n≥0

E[In(fn+1(·, x, t)1∆n) | Ft]X(dx, dt)

= E[F ] +

∫M×IR+

E[∇−x,tF | Ft]X(dx, dt)

The extension of this statement to F ∈ L2(Ω) is a consequence of the fact that the

adapted projection of ∇−F extends to a continuous operator from L2(Ω) into the

space of adapted processes in L2(Ω) ⊗ L2(M × IR+). For F =∑∞

n=0 In(fn) ∈ Sand u =

∑∞n=0 In(un+1) ∈ U with un+1 ∈ L2(IR+,K)n ⊗ L2(IR+,K), n ∈ IN, we can

extend a classical argument:

| E[∫

M×IR+

u(x, t)E[∇−x,tF | Ft]µ(dx)dt

]|

≤∞∑n=0

(n+ 1)! |∫M×IR+

(fn+1(·, x, t)1[0,t](·), un+1(·, t))L2(IR+,K)⊗nµ(dx)dt |

≤∞∑n=0

n!√n+ 1 ‖ fn+1 ‖L2(IR+,K)⊗(n+1)‖ un+1 ‖L2(IR+,K)⊗(n+1)

≤

(∞∑n=0

n! ‖ fn ‖2L2(IR+,K)⊗n

∞∑n=0

n! ‖ un+1 ‖2L2(IR+,K)⊗(n+1)

)1/2

≤ ‖ F ‖L2(Ω)‖ u ‖L2(Ω)⊗L2(M×IR+) .

2

26

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