Anticipative calculus with respect to filtered Poisson processes

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www.elsevier.com/locate/anihpb

Anticipative calculus with respect to filtered Poisson processes

L. Decreusefond

^a,b,^∗

, N. Savy

^c

aGET, Ecole Nationale Supérieure des Télécommunications, LTCI-UMR 5141, CNRS, 46, rue Barrault, 75634 Paris, France bENST – LTCI-UMR 5141 CNRS, 46, rue Barrault, 75634 Paris, France

cInstitut de Recherche Mathématiques de Rennes, Université de Rennes 1, 35042 Rennes cedex, France Received 5 March 2004; accepted 22 March 2005

Available online 18 November 2005

Abstract

We construct the basis of a stochastic calculus for a new class of processes: filtered Poisson processes. These processes are defined by an fBm-like stochastic integral but a Poisson process is subsided to the Brownian motion. We use Malliavin calculus to first construct a gradient then a divergence operator, which will play the role of an anticipative stochastic integral. We study into details the sample-paths regularity of this integral and give an Itô formula for Itô-like processes.

Résumé

Les processus de Poisson filtrés sont définis par une intégrale stochastique d’un noyau déterministe relativement à un processus de Poisson. Ils sont au processus de Poisson ce que le mouvement brownien fractionnaire est au mouvement brownien ordinaire.

Nous construisons pour ces processus les bases du calcul anticipatif. Ce travail similaire à celui d’Oksendal et ses collaborateurs sur les processus de sauts de Lévy, emprunte une autre approche que celle du calcul de Hida. Nous introduisons l’opérateur gradient sans utiliser la décomposition en chaos puis nous définissons son adjoint, ce qui nous permet de construire une notion d’intégrale anticipante. Les propriétés de régularité trajectorielle de cette intégrale sont précisées. Nous terminons notre travail par une formule d’Itô pour ces processus.

MSC:60H05; 60H07; 60G55

Keywords:Anticipative calculus; Malliavin calculus; Poisson process; Shot-noise process; Stochastic integral Mots-clés :Calcul anticipatif ; Calcul de Malliavin ; Processus de Poisson ; Processus shot-noise

1. Motivation

Modeling physical phenomena by stochastic processes should satisfy two constraints: a model must capture the prominent features of the phenomenon under consideration but it must also lead to tractable computations. It is thus necessary to always enlarge our catalog of processes for which we can do some calculations. Motivated by potential

* Corresponding author.

E-mail addresses:Laurent.Decreusefond@enst.fr (L. Decreusefond), Nicolas.Savy@univ-brest.fr (N. Savy).

doi:10.1016/j.anihpb.2005.03.005

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applications to mathematical finance, we are interested in stochastic calculus, anticipative or not, with respect to a new class of processes we call filtered Poisson processes. By filtered Poisson process (fpp for short), we mean a process defined by:

N_t^K= t

0

R^d

zK(t, s)ω(ds,dz), (1)

whereωis a marked Poisson measure, i.e., with a deterministic compensator (or dual predictable projection)ν and K a deterministic kernel. WhenK is of the formK(t, s)=k(t−s), N^K is a so-called Shot-Noise process. Such processes provide reasonable models for many phenomena in electronics, hydrology, climatology, telecommunications (see [14,22,23] and references therein), insurance (see [5] and references therein) and finance (see [4] and [21] for a review on this topic).

These processes are intimately related to pure jump Lévy processes. Pure jump Lévy processes have the representation

N_t= t

0

R

z

ω(ds,dz)−ν(ds,dz) ,

whereνis the compensator ofωand is assumed to be deterministic. TakingK(t, s)=1_[0,t](s)andd=1,a pure jump Lévy process thus appears to be a filtered Poisson process plus a deterministic process. When it comes to stochastic calculus, deterministic processes are in some sense negligible, thus what follows can also be applied to pure jump Lévy processes.

IfK,in (1), is smooth enough,N^K is a semi-martingale so that stochastic calculus with respect to this process can be done inside the classical framework. We want here to construct the tools required to handle the situation of a somehow singular kernelKand the tools required for anticipative calculus. This means that we want to develop a Malliavin calculus for our new class of processes. Such a problem has already been considered in several publications:

a first track consists of using chaotic decomposition as in [13,16] and more recently by [3,8,12] for pure jump Lévy processes. Developing and using Hida calculus for pure jump Lévy processes, Di Nunno et al. [8] constructed an anticipative integral and established, among other results, an anticipative Itô formula.

As usual when it comes to Malliavin calculus for jump processes, we can choose another approach and introduce a different gradient operator via a sample-path perturbation (see [1,16,6]). That is the way we proceeded here. Still, it must be noted that our definition of the gradient does not coincide with the definition given in [1] or [16]. Actually, our goal is to define a gradient such that its adjoint restricted to predictable processes coincides with the Lebesgue–

Stieltjes integral with respect toω−ν.For, we have to “twist” the perturbation in the definition ofDF (h), see (2), otherwise we would have additional terms as in [1, Formula 3.11]. The paper is organized as follows. In Section 2, we construct the basis of the Malliavin calculus for marked point processes. The material here is mainly excerpted of [6] though we can establish more properties because of the specific form of the compensator. Once this is done, we develop the stochastic calculus with respect toN^Kin Section 3. SinceN^Kmay have finite variation, we take care to compare Lebesgue–Stieltjes integral with our newly constructed integral. As Theorem 19 shows, they do not coincide.

Nevertheless, the divergence is an interesting tool because it is the convenient one to establish the Itô formula by which we conclude this paper. The necessary material about deterministic fractional calculus is recalled in Appendix A.

2. Stochastic calculus for marked point processes 2.1. Introduction

The space of simple, locally finite on[0, T] ×R^d(T deterministic finite or not) integer-valued measure is denoted byΩ. Assume that we are given a probability measureηonE=R^d and a positive, square integrable function,λ, on[0, T] (λwill both denote the function s→λ(s) and the measureλof densityλ with respect to the Lebesgue measure) such thatm:=infs∈[0,T]λ(s) >0.We define the probabilityPas the unique measure onΩ such that the

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canonical measureωis a Poisson random measure of compensatorν(ds,dz):=λ(s)ds η(dz). The canonical filtration Fis defined by:

F0= {∅, Ω} and Ft=σ s

0

B

ω(ds,dz), st, B∈B(R^d)

.

The predictableσ-algebra onΩ×R⁺×R^dis denoted byP. We denote byT_nthen-th jump time and byZ_nthen-th mark, this means that

ω(ds,dz)=

n1

δ_T_n_,Z_n(s, z).

The mark point processN is defined by N_t=

n∈N^∗∗

Z_nI[T_nt].

Following [10], we introduce the following notations: For every measurable and locally bounded or non-negative processf,

(f∗µ)_t(ω):=

t

0

R^d

f (s, z)(ω)µ(ω,ds,dz)=

n1

f

T_n(ω), Z_n(ω)

I_[Tn(ω)t].

The processN is of finite variation on every compact of time. Thus, for every measurable and locally bounded or non-negative processXthe process defined by:

X^(SL)^∗ N

t(ω):=

t

0

X_s(ω)dNs(ω)

exists for a.e.ω∈Ω andt∈R⁺in Stieltjes–Lebesgue way and we have:

X^(SL)^∗ N

t(ω)=

n1

Zn(ω)XT_n(ω)(ω)I[T_n(ω)t]. We recall here the main results of [6].

Definition 1.A functional is said to be cylindric whenever it is of the form F =f

T 0

R^d

f1(s)g1(z)ω(ds,dz), . . . , T

0

R^d

f_n(s)g_n(z)ω(ds,dz)

,

wheref is a bounded twice differentiable function with bounded derivatives,f_ig_i∈L²(ν) f_i is continuously differentiable with bounded derivatives for alli=1, . . . , n. We denote bySthe set of cylindric functionals.

Definition 2.For any functionalsF∈Sand anyh∈L²(ν),we defineDF (h)by DF (h)= −

n

i=1

∂f

∂x_i T

0

R^d

f₁(s)g₁(z)ω(ds,dz), . . . , T

0

R^d

f_n(s)g_n(z)ω(ds,dz)

× T

0

R^d

f_i(s)gi(z)

1 λ(s)

s

0

h(r, z)λ(r)dr

ω(ds,dz). (2)

The main properties ofDF (h)are summarized in the following theorem:

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Theorem 1.

(1) Dis a derivation:forF,G∈S, andh∈L²(ν), DF G(h)=F DG(h)+GDF (h).

(2) Dsatisfies the following integration by parts formula:

E DF (h)

=E F ·

h^(SI)^∗ (ω−ν)

T

. (3)

The particular hypothesis we have here, enables us to simplify the theory developed in [6].

Theorem 2.For allF∈Sthere exists a constantc >0such that, for anyh∈L²(ν), we have:

E DF (h)²

ch²_L2(ν).

In what follows,cdenote any irrelevant constant and may vary from line to line.

Proof. Using the boundedness of the functionsf_i,f_iandg_i there exists a constantcsuch that:

E DF (h)²

cE

T

0

R^d

1 λ(s)

s

0

h(r, z)λ(r)drω(ds,dz)

2 .

Sinceλis lower-bounded bym >0, we have:

E DF (h)² cE

T 0

R^d

s 0

h(r, z)λ(r)dr

ω(ds,dz)

2 . Moreover, using Cauchy–Schwarz inequality,

E T

0

R^d

h(s, z) ω(ds,dz)

2

2E

T

0

R^d

h(s, z)(ω−ν)(ds,dz)

2 +2

T

0

R^d

h(s, z)ν(ds,dz)

2

2E

T 0

R^d

h(s, z)²(ω−ν)(ds,dz)

E T

0

R^d

(ω−ν)(ds,dz)

+2 T

0

R^d

h(s, z)²ν(ds,dz)

2E

T 0

R^d

ν

[0, T] ×E +2

T

0

R^d

2

1+ν

[0, T] ×R^d^T

0

R^d

h(s, z)²ν(ds,dz).

We thus get the following relation:

E DF (h)²

c

T

0

R^d

s 0

h(r, z)λ(r)dr 2

ν(ds,dz). (4)

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Using the special form of the measureν, we can write:

T

0

R^d

s 0

h(r, z)λ(r)dr 2

λ(s)η(dz)ds T

0

R^d

s 0

h²(r, z)λ(r)dr s

0

λ²(r)dr

λ(s)η(dz)ds

T

0

R^d

T 0

h²(r, z)λ(r)dr T

0

λ²(r)dr

λ(s)η(dz)ds

λ²₂λ1

T

0

R^d

h²(r, z)λ(r)drη(dz).

From that and from relation (4), we infer that E DF (h)²

c

T

0

R^d

h²(r, z)λ(r)drη(dz).

The proof is thus complete. 2

Theorem 3. For anyF ∈S, there exists∇F ∈L²([0, T] ×E×Ω, ν⊗dP)measurable with respect to the three variables such that:

DF (h)= ∇F, h_L²(ν) for allh∈L²(ν) Proof. LetF∈Sbe fixed. Consider the application:

φ_F:L²(ν)⊗L²(Ω)→R,

h⊗G→E GDF (h) .

It is clearly linear, let us show its continuity. One considers the projective tensor product (see Köthe [11] Section 41 for details) equipped with the following semi-norm:

z =inf n

i=1

h_i_L²(ν)G_i2

where the ‘inf’ is taken over all the possible decompositions ofzin the formz=n

i=1hi⊗Gi. It is then sufficient to show the existence of a constantcsuch that:

φ_F

_n

i=1

h_i⊗G_i c

n

i=1

h_i⊗G_i

. (5) According to Hölder inequality and to the previous theorem, we have

φF

_n

i=1

hi⊗Gi

n

i=1

E GiDF (hi) n

i=1

Gi2DF (hi)

2

c

n

i=1

G_i2h_i_L²(ν)c

n

i=1

h_i⊗G_i .

Thus,φF is a continuous linear form. Using the identification of tensor product ofL²type spaces:

L²

Ω× [0, T] ×R^d;dP⊗v

∼L²(ν)⊗L²(Ω),

we can invoke the Riesz representation theorem which ensures the existence of a measurable random variable∇F∈ L²([0, T] ×R^d×Ω, ν⊗dP)such that:

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φ_F(G⊗h)= ∇F;G⊗h_L²(Ω)⊗L²(ν)

=E G ∇F;h_L²(ν)

=EGDF (h) . The proof is thus complete. 2

Theorem 4.The applicationF → ∇F is closable inL²(P).

Proof. Let{F_n, n1}a sequence ofSsuch thatF_ntends to 0 inL²(P)and such that∇F_ntends to a limit calledL.

For anyG∈Sand anyh∈L²(ν)we have:

E L, h ·G

= lim

n→∞E ∇F_n, h_L²(ν)·G

= lim

n→∞E ∇F_n·G, h_L²_(ν)

= lim

n→∞

E ∇(F_nG), h

L²(ν)

−E ∇G·F_n, h_L²(ν)

= lim

n→∞

E ∇(FnG), h

L²(ν)

−E ∇G, h_L²(ν)Fn

= lim

n→∞

EFnG·

h^(SI)^∗ (ω−ν)

T

−E ∇G, h_L²(ν)Fn

=0,

sinceF_ntends to 0 inL²(P). AsS is dense inL²(P), we infer thatL=0P-a.e. 2 Definition 3.OnS, we introduce the norm: for anyF∈S,

F²_2,1= F²₂+E ∇F²_L2(ν)

,

and the spaceD2,1, the closure ofSwith respect to this norm.

The usual properties then follow (see [6]): for(F, G)∈(D2,1)², andφ∈C_b¹, we have:

∇F G=G∇F +F∇G and ∇ φ(F )

=φ(F )∇F.

Definition 4.Letζ be anL²(ν)-valued random variable, it is in the domain ofδ(dom(δ))iff, there existscsuch that for anyF ∈Swe have:

EDF (ζ )cF2. In this case,δ(ζ )is defined by:

E F δ(ζ )

=EDF (ζ ) .

Proposition 1.Any element ofL²(ν)belongs todom(δ)andδ(h)=(h^(SI)^∗ (ω−ν))_T. Proof. Since, forh∈L²(ν), we have:

E DF (h)

=E F·

h^(SI)^∗ (ω−ν)

T

, (6)

it is clear thatL²(ν)⊂dom(δ)and by identification thatδ(h)=(h^(SI)^∗ (ω−ν))_T. 2 Theorem 5.For anya∈Sand anyh∈L²(ν),a·hbelongs todom(δ)and

δ(ah)=aδ(h)−Da(h). (7)

Proof. ForF∈S, it is easy to see, by definition ofDF (h), that ifais a random variable, we have:

DF (ah)=aDF (h).

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Moreover, sinceSis an algebra, we get:

E DF (ah)

=EaDF (h)

=E D(aF )(h)−F D(a)(h)

=EF aδ(h)

−E F D(a)(h) . It is then clear that:

E DF (aζ )cF2,

this shows thata·h∈dom(δ)and thatδ(ah)=aδ(h)−Da(h). 2

Now, we generalize the previous construction by consideringL²(ν)-valued random variable. The set of cylindrical processesS(L²(ν))is the set of processes of the form:

Φ= k

i=1

Fivi, Fi∈S, vi∈L²(ν).

By definition, DΦ(h)=

k

i=1

DF_i(h)⊗v_i, F_i∈S, v_i∈L²(ν).

Proposition 2.For anyΦ∈S(L²(ν)),∇Φ is an Hilbert–Schmidt operator.

Proof. Note first that the mapf →.

0f (r)λ(r)dr is Hilbert–Schmidt from L²(ds) into itself. Indeed, this can be written

T

0

1_[0,t](r)λ(r)f (r)dr,

and the kernel(t, r)→1_[0,t](r)λ(r)is clearly square integrable on[0,1]².

Let(a_j, j1)and(b_j, j 1)be CONB of respectivelyL²(λ)andL²(η). It is clear that E

_∞

j,k=1

DΦ(a_j⊗b_k);DΦ(a_j⊗b_k)

L²(ν)

cE

_∞

j,k=1

T 0

1 λ(s)

s

0

a_j(r)b_k(z)λ(r)dr ω(ds,dz) 2

cE _∞

j=1

T

0

s 0

a_j(r)λ(r)dr 2

ds

, and the result follows. 2

To motivate the following, recall that in the Brownian setting, it is well known that E δ(u)²

=E u²

+E trace(∇u◦ ∇u)

. (8)

Unfortunately, this property does not hold any longer in our situation. We want to find its analog. On one hand, one can keep the analog of the second term of (8) but we have to introduce an operatorΓ such that:

E δ(u)²

=E u, Γ u_L²(ν)

+E trace(∇u◦ ∇u) .

This is the main result of Theorem 6. On the other hand, one can keep the analog of the first term of (8) but we have to define another gradient ˆ∇such that:

E δ(u)²

=E u²_L2(λ)

+E trace(ˆ∇u◦ ˆ∇u) .

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This is the result of Theorem 10, which is valid only when one does not have marks. The gradient ˆ∇is easily linked with the covariant derivative introduced by Privault in [15,17–19].

We need to introduce a few notations. Consider the following spaceH^ν: H^ν=

h∈L²(ν): ∂h

∂s ∈L²(ν)

equipped with the following inner product:

g, h_H^ν = g, h_L²(ν)+ ∂g

∂s,∂h

∂s

L²(ν)

.

Consider{ε_i: i∈N^∗}a CONB ofH^ν. ObviouslyH^ν⊂L²(ν)and define with obvious notationsS(H^ν).

Proposition 3.Let us consider the following operator:

Γ:S H^ν

→L²

Ω× [0, T] ×E,dP⊗ν , ζ→^∞

i=1

ζ, ε_i_H^νΓ (ε_i) withΓ (εi)= ∇(δ(εi)). Then,Γ is continuous.

Proof. Let us show that for anyh∈S(H^ν), E ∇δ(h)²

L²(ν)

<∞. First of all, leth∈H^ν, we have:

∇δ(h)²_L₂

(ν)= ∞ i=1

∇ δ(h)

, e_i2 L²(ν)=

∞ i=1

D δ(h)

(e_i)². Butδ(h)is cylindrical, so we can write:

E ∇δ(h)²

L²(ν)

^∞

i=1

E T

0

R^d

∂h

∂s(s, z) 1 λ(s)

s

0

e_i(r, z)λ(r)dr ω(ds,dz) 2

∞ i=1

T

0

R^d

∂h

∂s(s, z) 1 λ(s)

s

0

ei(r, z)λ(r)dr 2

ν(ds,dz)

T

0

R^d

∂h

∂s

2

(s, z) 1 λ(s)²

∞ i=1

ei,I_[0,T]²_L2(ν)

ν(ds,dz)

1

m² T

0

R^d

∂h

∂s

2

(s, z)I_[0,T]²_L2(ν)ν(ds,dz) 1 m²

T

0

R^d

∂h

∂s

2

(s, z)ν(ds,dz)

1

m² ∂h

∂s ²

L²(ν)

. We thus have:

δ(h)

2,1=E δ(h)²

+E ∇δ(h)²

L²(ν)

h²_L2(ν)+ 1 m²

∂h

∂s ²

L²(ν)

1∧ 1 m²

h²_H^ν.

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δ:H^ν→D2,1 is thus continuous and∇:D2,1→L²(Ω× [0, T] ×E,dP⊗ν)is also continuous by construction.

Then,

∇ ◦δ:H^ν→L²

Ω× [0, T] ×E,dP⊗ν

is a linear continuous application, i.e., there exists a constantcsuch that, for anyh∈H^ν: E ∇δ(h)²_L₂

(ν)

ch²_H^ν (9) and thus for anyi∈N^∗

E ∇

δ(εi)²_L₂

(ν)

c. (10) Consider nowζ ∈S(H^ν),ζ=F hwithF ∈S(thus bounded) andh∈H^ν. Then, we have:

E Γ (ζ )²

L²(ν)

=E

∞ i=1

ζ, ε_i_H^νΓ (ε_i) L²(ν)

2

F²_∞^∞

i=1

h, ε_i²_H^νE Γ (ε_i)

L²(ν)

2

cF²_∞^∞

i=1

h, ε_i²_H^νcF²_∞h²_H^ν <∞. The proof is thus complete. 2

Theorem 6.

(1) For anyζ∈S(H^ν)we have:

δ(ζ )=^∞

i=1

ζ, ε_i_H^νδ(ε_i)−

∇.ζ, ε_i_H^ν, ε_i(·)

L²(ν)

. (11)

(2) Forζ ∈S(H^ν)we have:

∇_∗δ(ζ )=^∞

i=1

ζ, ε_i_H^ν∇_∗ δ(ε_i)

+δ(∇_∗ζ ).

∗indicates the free variable.

(3) For anyζ∈S(H^ν)we have:

Eδ(ζ )²

=E ζ;Γ ζ_L²(ν)

+E trace(∇ζ ◦ ∇ζ )

. (12)

Proof. To prove the first point, we reproduce the proof of Theorem 5. Furthermore, by linearity and continuity ofδ: δ(ζ )=^∞

i=1

δ

ζ;ε_i_H^νε_i .

Applying the first point of this theorem witha= ζ;ε_i_H^ν we get:

δ(ζ )=^∞

i=1

ζ;ε_i_H^νδ(ε_i)−

∇.

ζ;ε_i_H^ν , ε_i(·)

L²(ν)

.

Sinceε_i is deterministic, we can invert the inner product and the gradient and write:

∇.

ζ;ε_i_H^ν , ε_i(·)

L²(ν)=

∇.ζ, ε_i_H^ν, ε_i(·)

L²(ν), so that the result holds.

On the one hand we know that:

δ(ζ )=^∞

i=1

ζ, ε_i_H^νδ(ε_i)−

∇.ζ, ε_i_H^ν, ε_i(·)

L²(ν)

. (13)

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On the other hand:

δ(∇∗ζ )= ∞

i=1

∇∗ζ, ε_i_H^νδ(ε_i)−

∇.(∇∗)ζ_., ε_i

H^ν, ε_i(·)

L²(ν)

. (14)

Applying∇ to both sides of (13), we get:

∇_∗ δ(ζ )

=^∞

i=1

ζ, ε_i_H^ν∇_∗ δ(ε_i)

+ ∇_∗

ζ, ε_i_H^ν

δ(ε_i)− ∇_∗

∇ζ, ε_i_H^ν, ε_i

L²(ν)

=^∞

i=1

∇_∗ζ, ε_i_H^νδ(ε_i)−

∇_∗ ∇ζ_., ε_i_H^ν, ε_i(·)

L²(ν)

+^∞

i=1

ζ, ε_i_H^ν∇_∗ δ(ε_i)

=^∞

i=1

∇_∗ζ, ε_i_H^νδ(ε_i)−

∇_∗∇ζ_., ε_i_H^ν, ε_i(·)

L²(ν)

+^∞

i=1

ζ, ε_i_H^ν∇_∗ δ(ε_i)

. According to (14), we get the second formula.

We are now in position to compute the second moment ofδ(ζ ). According to the previous results, we have:

E δ(ζ )²

=E δ(ζ )δ(ζ )

=E ζ_∗,∇_∗δ(ζ )

L²(ν)

=E

ζ_∗, ∞ i=1

ζ, ε_i_H^ν∇∗ δ(ε_i)

L²(ν)

+E ζ_∗, δ

∇∗(ζ )

L²(ν)

=E ζ_∗, Γ ζ_L²(ν)

+E ζ_∗, δ

∇_∗(ζ )

L²(ν)

. Furthermore,

E ζ_∗, δ

∇_∗(ζ )

L²(ν)

=E T

0

R^d

ζs,zδ(∇s,zζ )ν(ds,dz)

=E T

0

R^d

T

0

R^d

∇u,vζs,z∇s,zζu,vν(du,dv)ν(ds,dz)

=E trace(∇ζ◦ ∇ζ ) , the proof is thus complete. 2

Consider, forΦ∈S(H^ν)the following norm:

Φ^Γ_2,12

=1 2

E Φ²_L2(ν)

+E Γ Φ²_L2(ν)

+E ∇Φ²_L2(ν)⊗L²(ν)

.

ConsiderD^Γ_2,1(H^ν)the closure ofS(H^ν)with respect to this norm.

Remark 1.In the Brownian framework,Γ is equal to identity and we get the usual normD2,1. Proposition 4.We have:

D^Γ2,1(H^ν)⊂dom(δ).

Proof. Considerζ ∈D^Γ_2,1(H^ν)and{ζ_n: n∈N^∗}a sequence ofS(H^ν)which tends toζ inD2,1(H^ν). For anyF ∈S, we have:

EDF (ζn)=E F δ(ζn)δ(ζn)

2F2ζn^Γ2,1F2, according to (12) and the result follows. 2

We now define the integral with respect toN.

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Definition 5.For anyufrom[0,1]intoR, we setu(s, z)˜ =zu(s). Ifu˜∈dom(δ), we defineδ^N(u)by δ^N(u)=δ(u),˜

andλ˜ is denoting the measure:

λ(ds)˜ =

R^d

z²ν(ds,dz)=

R^d

z²η(dz)λ(s)ds.

Note that by construction,u_L2(˜λ)= ˜u_L²(ν).

Proposition 5.For anyh∈L²(˜λ),hbelongs todom(δ^N)and δ^N(h)=

h^(SL)^∗ N

T.

Proof. Sinceδcoincides with the Stieltjes–Lebesgue integral with respect to(ω−ν)for predictable processes, we have:

δ^N(u)= T

0

R^d

zu(s)(ω−ν)(ds,dz)= u^(SL)^∗ N

T. 2

We denote byD^Nthe adjoint ofδ^N. Proposition 6.We have

D^NF (u)=DF (u)˜ for anyF ∈S. The associated gradient is equal to:

∇^NF = 1

R^dz²η(dz)

R^d

∇s,zF zη(dz) for anyF ∈S.

Proof. The first point comes from the following identity:

E F δ^N(u)

=E F δ(u)˜

=E DF (u)˜ . For anyF ∈S, we have:

D^NF (u)= T

0

R^d

∇s,zF zη(dz)u(s)λ(s)ds= T

0

1

R^dz²η(dz)

R^d

∇s,zF zη(dz)u(s)λ(ds)˜

= ∇^NF, u_L2(˜λ). 2

Following the same lines as above, we can introduce, on the space of cylindrical processes, a norm · ^N_2,1 and extend this space with respect to this norm to get a space denoted byD^N_2,1 (H^λ^˜). We can easily prove the next theorem.

Theorem 7.

(1) D^N_2,1 (H^λ^˜)⊂dom(δ^N).

(2) For anya∈D2,1and anyζ∈D^N_2,1 (H^λ^˜)we have:

δ^N(aζ )=aδ^N(ζ )− ∇^Na;ζ_L²_(˜_λ).

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(3) For anyζ ∈D^N_2,1 (H^λ^˜)we have:

E δ^N(ζ )²

=E ζ;Γ^Nζ_L²_(˜_λ)

+Etrace(∇^Nζ◦ ∇^Nζ ) .

ThusD^N_2,1 (H^λ^˜)is included in theδ^Ndomain. We now state the relationships between any of the integrals previously defined.

Theorem 8.Letu∈D2,1(L²(ν)), then we have:

δ(u)=

u^(SI)^∗ (ω−ν)

T − T

0

R^d

∇s,zus,zν(ds,dz).

Furthermore, foru∈D^N_2,1 (L²(˜λ)), we have:

δ^N(u)= u^(SL)^∗ N

T − T

0

∇s^Nusλ(ds).˜

Proof. For the sake of clarity, we will prove only the second point. The first one is analogous. Letu=_n

i=1uiI(t_i,t_i₊₁]

withui ∈D2,1 and let φ∈S. Since the Skorohod integral and the Stieltjes–Lebesgue coincides on deterministic processes:

E u^(SL)^∗ N

·φ

=E _n

i=1

u_i I(t_i,t_i+1]

(SL)∗ N φ

= n

i=1

Eδ^N(I(t_i,t_i+1])u_iφ

= n

i=1

E ∇^N[u_iφ];I(t_i,t_i+1])

L²(˜λ)

= n

i=1

E φ∇^Nu_i;I(t_i,t_i₊₁]_L²₍_λ)_˜ +

n

i=1

E u_i∇^Nφ;I(t_i,t_i₊₁]_L²₍_λ)_˜

=A₁+A₂. Moreover,

A₁=E

φ T

0

∇s^N

_n

i=1

u_iI(t_i,t_i+1](s)

λ(ds)˜

=E

φ T

0

∇s^Nu_sλ(ds)˜

,

and

A2=E

∇^Nφ; n

i=1

uiI(t_i,t_i₊₁]

L²(˜λ)

=E

φδ^N _n

i=1

uiI(t_i,t_i₊₁]

.

The equality is true for anyφ, so, the result is established for processes of the formu=n

i=1uiI(ti,ti+1]. The general result follows by a limit procedure. 2

Note that∇_s^Nu_s is equal to 0 dP⊗ ˜λ(ds)a.s. forupredictable and inD2,1(L²(ν)), it is then straightforward that Theorem 9.The integralsδand^(SI)_∗ coincide for predictable processes and the integralsδ^N,^(SL)_∗ and^(SI)_∗ coincide for predictable processes.

2.2. Covariant derivative and Weitzenböck formula Consider the space

H^λ^∗=

h∈L²(λ): h∈L²(λ), h(T )=0 ,

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wherehdenotes the time derivative ofh, equipped with the scalar product g, h_H^λ∗= g, h_L²(λ)+ g, h_L²(λ).

Introduce the following operators:

I¹:L² → H^λ^∗, u →

t→

t

0

u(s)ds

and I⁻¹:H^λ^∗ → L², u → u.

Definition 6.For(u, v)∈ [H^λ^∗]², we define:

˜∇u(v)= −1

λI⁻¹[v]I¹[λu]. (15)

We can write

˜∇u(v)=

˜∇.(v), u(·)

L²(λ)

where

˜∇s(v):t→ ˜∇s(v)(t )= − 1

λ(t )I⁻¹[v](t )I_[0,t](s). (16)

Proposition 7.Let(u, v)∈ [H^λ^∗]². We have:

D δ(v)

(u)=δ

˜∇u(v)

+ u, v_L²(λ). (17)

Proof. Considerq_u:s→ _λ(s)¹ _s

0u(r)λ(r)dr. We havevq_u= − ˜∇u(v)and by the very definition ofD:

D δ(v)

(u)= T

0

˜∇u(v)(s)ω(ds)=

˜∇u(v)^(SI)^∗ (ω−λ)

T − T

0

v(s)q_u(s)λ(s)ds

=δ

˜∇u(v)

−

v(s)q_u(s)λ(s)T 0 −

T

0

v(s)[q_uλ](s)ds

=δ

˜∇u(v)

+ u, v_L²(λ). 2

Proposition 8.Consider nowF ∈Sand(u, v)∈ [H^λ^∗]². We have:

D DF (u)

(v)−D DF (v)

(u)=DF

˜∇^v(u)− ˜∇u(v) . The proof rests on the following lemma.

Lemma 1.Let(u, v)∈ [H^λ^∗]², then

I⁻¹[q_u]q_v−I⁻¹[q_v]q_u=q_w, (18)

with w=1

λ

I¹[λv]I⁻¹[u] −I¹[λu]I⁻¹[v]

= ˜∇u(v)− ˜∇v(u). (19)

Proof. On the one hand, as I⁻¹[q_u] = −I⁻¹[λ]

λ² I¹[λu] +u, we have: