Integration by parts formula and applications to equations with jumps

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Integration by parts formula and applications to equations with jumps

Emmanuelle Clement, Vlad Bally

To cite this version:

Emmanuelle Clement, Vlad Bally. Integration by parts formula and applications to equations with jumps. Probability Theory and Related Fields, Springer Verlag, 2011, 151 (3-4), pp.613-657. �hal- 00431632�

Integration by parts formula and applications to equations with jumps

Vlad Bally^∗ Emmanuelle Cl´ement^∗ Preliminary version, November 2009

Abstract

We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes solution of stochastic differential equations with jumps, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bismut and Bichteler, Gravereaux and Jacod fails.

2000 MSC. Primary: 60H07, Secondary 60G51

Key words: Integration by parts formula, Malliavin Calculus, Stochastic Equations, Poisson Point Measures.

1 Introduction

This paper is made up of two parts. In a first part we give an abstract, finite dimensional version of Malliavin calculus. Of course Malliavin calculus is known to be an infinite dimensional differential calculus and so a finite dimensional version seems to be of a limited interest. We discuss later on the relation between the finite dimensional and the infinite dimensional frameworks and we highlight the interest of the finite dimensional approach.

In the second part of the paper we use the results from the first section in order to give sufficient conditions for the regularity of the law ofX_t,whereXis the Markov process with infinitesimal operator

Lf(x) =h∇f(x), g(x)i+ Z

R^d

(f(x+c(z, x))−f(x))γ(z, x)µ(dz). (1) Suppose for the moment that γ does not depend on x. Then it is well known that the process X may be represented as the solution of a stochastic equation driven by a Poisson point measure with

∗Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UMR 8050, Universit´e Paris-Est Marne-la-Vall´ee, 5 Bld Descartes, Champs-sur-marne, 77454 Marne-la-Vall´ee Cedex 2, France.

intensity measureγ(z)µ(dz).Sufficient conditions for the regularity of the law ofX_t using a Malliavin calculus for Poisson point measures are given in [B.G.J].But in our frameworkγ depends onxwhich roughly speaking means that the law of the jumps depends on the position of the particle when the jump occurs. Such processes are of interest in a lot of applications and unfortunately the standard Malliavin calculus developed in [B.G.J] does not apply in this framework. After the classical paper of Bichteler Gravereaux and Jacod a huge work concerning the Malliavin calculus for Poisson point measures has been done and many different approaches have been developed. But as long as we know they do not lead to a solution for our problem. If X is an one dimensional process an analytical argument permits to solve the above problem , this is done in [F.1],[F.2] and [F.G] but the argument there seems difficult to extend in the multi-dimensional case.

We come now back to the relation between the finite dimensional and the infinite dimensional framework. This seems to be the more interesting point in our approach so we try to explain the main idea. In order to prove Malliavin’s regularity criterion for the law of a functional F on the Wiener space the main tool is the integration by parts formula

E(∂^βf(F)) =E(f(F)H_β) (2)

where∂^β denotes the derivative corresponding to a multi-index β and H_β is a random variable built using the Malliavin derivatives ofF.Once such a formula is proved one may estimate the Fourier trans- form bp_F(ξ) = E(exp(iξF)) in the following way. First we remark that ∂_x^βexp(iξx) = (iξ)^βexp(iξx) (with an obvious abuse of notation) and then, using the integration by parts formula

|bp_F(ξ)| = 1

|ξ|^|β|

E(∂_x^βexp(iξF))

= 1

|ξ|^|β||E(exp(iξF)H_β)| ≤ 1

|ξ|^|β|E|H_β|.

If we know thatE|H_β|<∞for every multi-indexβ then we have proved that|ξ|^p|bp_F(ξ)|is integrable for every p∈N and consequently the law of F is absolutely continuous with respect to the Lebesgue measure and has an infinitely differentiable density.

Let us come back to the infinite dimensional differential calculus which permits to built H_β. In order to define the Malliavin derivative of F one considers a sequence of simple functionals F_n→ F in L² and, if DF_n →G inL²,then one defines DF =G. The simple functionals F_n are functions of a finite number of random variables (increments of the Brownian motion) and the derivative DF_n is a gradient type operator defined in an elementary way. Then one may take the following alternative

way in order to prove the regularity of the law of F. For each fixed n one proves the analogues of the integration by parts formula (2): E(∂^βf(F_n)) =E(f(F_n)H_βⁿ).AsF_nis a function which depends on a finite number m of random variables, such a formula is obtained using standard integration by parts on R^m (this is done in the first section of this paper). Then the same calculus as above gives

|bp_Fn(ξ)| ≤ |ξ|^−|β|E H_βⁿ

.Passing to the limit one obtains

|bp_F(ξ)|= lim

n |bp_Fn(ξ)| ≤ |ξ|^−|β|sup

n

EH_βⁿ and, if we can prove that sup_nE

H_βⁿ

< ∞, we are done. Notice that here we do not need that Fn → F in L² but only in law. And also, we do not need to built H_β but only to prove that sup_nE

H_βⁿ

<∞.Anyway we are not very far from the standard Malliavin calculus. Things become different if sup_nE

H_βⁿ

= ∞ and this is the case in our examples (because the Ornstein Uhlenbeck operators LF_n blow up as n → ∞). But even in this case one may obtain estimates of the Fourier transform of F in the following way. One writes

|bp_F(ξ)| ≤ |bp_F(ξ)−pb_Fn(ξ)|+|bp_Fn(ξ)| ≤ |ξ| ×E|F −F_n|+|ξ|^−|β|E H_βⁿ

.

And if one may obtain a good balance between the convergence to zero of the error E|F−F_n| and the blow up to infinity ofE

H_βⁿ

then one obtains|bpF(ξ)| ≤ |ξ|^−p for somep. Examples in which such a balance works are given in Section 3. An other application of this methodology is given in [B.F]

for the Boltzmann equation. In this case some specific and nontrivial difficulties appear due to the singularity and unboundedness of the coefficients of the equation.

The paper is organized as follows. In Section 2 we establish the abstract Malliavin calculus associ- ated to a finite dimensional random variable and we obtain estimates of the weight H_β which appear in the integration by parts formula (we follow here some ideas which already appear in [B],[B.B.M]

and [Ba.M]). Section 3 is devoted to the study of the regularity of the law of the Markov processX of infinitesimal operator (1) and it contains our main results : Proposition 3 and Theorem 4. At last we provide in Section 4 the technical estimates which are needed to prove the results of section 3.

2 Integration by parts formula

2.1 Notations-derivative operators

Throughout this paper, we consider a sequence of random variables (V_i)_i∈N∗ on a probability space (Ω,F, P), a sub σ-algebra G ⊆ F and a random variable J, G measurable, with values in N. We

assume that the variables (V_i) and J satisfy the following integrability conditions : for all p ≥ 1, E(J^p) +E((PJ

i=1V_i²)^p) < ∞. Our aim is to establish a differential calculus based on the variables (V_i), conditionally on G, and we first define the class of functions on which this differential calculus will apply. More precisely, we consider in this paper functionsf : Ω×R^N∗→Rwhich can be written as

f(ω, v) = X∞

j=1

f^j(ω, v1, ..., vj)1_{J(ω)=j} (3) wheref^j : Ω×R^j →R areG × B(R^j)−measurable functions. We denote by Mthe class of functions f given by (3) such that there exists a random variable C ∈ ∩q≥1L^q(Ω,G, P) and a real number p ≥ 1 satisfying |f(ω, v)| ≤ C(ω)(1 + (PJ(ω)

i=1 v_i²)^p). So conditionally onG, the functions of M have polynomial growth with respect to the variables (V_i). We need some more notations. Let Gi be the σ−algebra generated by G ∪σ(V_j,1 ≤ j ≤ J, j 6= i) and let (a_i(ω)) and (b_i(ω)) be sequences of Gi

measurable random variables satisfying−∞ ≤a_i(ω)< b_i(ω)≤+∞, for alli∈N^∗. Now letO_i be the open set of R^N∗ defined byO_i =P_i⁻¹(]a_i, b_i[), whereP_i is the coordinate mapP_i(v) =v_i. We localize the differential calculus on the sets (Oi) by introducing some weights (πi), satisfying the following hypothesis.

H0. For all i∈N^∗,πi ∈ M, 0≤πi ≤1 and{πi>0} ⊂Oi. Moreover for allj≥1,π_i^j is infinitely differentiable with bounded derivatives with respect to the variables (v₁, . . . , v_j).

We associate to these weights (πi), the spaces C_π^k ⊂ M, k ∈ N^∗ defined recursively as follows.

For k = 1, C_π¹ denotes the space of functions f ∈ M such that for each i∈ N^∗, f admits a partial derivative with respect to the variable vi on the open set Oi. We then define

∂_i^πf(ω, v) :=π_i(ω, v) ∂

∂vi

f(ω, v) and we assume that ∂^π_if ∈ M.

Note that the chain rule is verified : for eachφ∈C¹(R^d,R) and f = (f¹, ..., f^d)∈(C_π¹)^d we have

∂_i^πφ(f) = Xd

r=1

∂_rφ(f)∂_i^πf^r.

Suppose now thatC_π^kis already defined. For a multi-indexα= (α1, ..., α_k)∈N^∗kwe define recursively

∂^π_α = ∂_α^π_k...∂_α^π₁ and C_π^k+1 is the space of functions f ∈ C_π^k such that for every multi-index α = (α₁, ..., α_k)∈N^∗k we have ∂^π_αf ∈C_π¹.Note that iff ∈C_π^k,∂_α^πf ∈ Mfor each α with|α| ≤k.

Finally we define C_π^∞ = ∩k∈N∗C_π^k. Roughly speaking the space C_π^∞ is the analogue of C^∞ with partial derivatives∂_i replaced by localized derivatives ∂_i^π.

Simple functionals. A random variable F is called a simple functional if there exists f ∈ C_π^∞ such that F =f(ω, V), where V = (V_i). We denote by S the space of the simple functionals. Notice thatS is an algebra. It is worth to remark that conditionally on G,F =f^J(V₁, . . . , V_J).

Simple processes. A simple process is a sequence of random variables U = (Ui)i∈N∗ such that for each i∈N^∗, U_i ∈ S. Consequently, conditionally on G, we have U_i =u^J_i(V₁, . . . , V_J). We denote by P the space of the simple processes and we define the scalar product

hU, ViJ = XJ

i=1

UiVi. Note thathU, ViJ ∈ S.

We can now define the derivative operator and state the integration by parts formula.

The derivative operator. We defineD:S → P : by

DF := (D_iF)∈ P whereD_iF :=∂_i^πf(ω, V).

Note thatDiF = 0 fori > J. For F = (F¹, ..., F^d)∈ S^d the Malliavin covariance matrix is defined by σ^k,k′(F) =D

DF^k, DF^k′E

J = XJ

j=1

D_jF^kD_jF^k′. We denote

Λ(F) ={detσ(F)6= 0} and γ(F)(ω) =σ⁻¹(F)(ω), ω ∈Λ(F).

In order to derive an integration by parts formula, we need some additional assumptions on the random variables (Vi). The main hypothesis is that conditionally onG,the law of the vector(V1, ..., VJ) admits a locally smooth density with respect to the Lebesgue measure onR^J.

H1. i) Conditionally on G, the vector (V1, ..., VJ) is absolutely continuous with respect to the Lebesgue measure onR^J and we note p_J the conditional density.

ii) The set{pJ >0}is open inR^J and on{pJ >0} lnpJ ∈C_π^∞. iii) ∀q≥1, there exists a constant C_q such that

(1 +|v|^q)p_J ≤C_q where|v|stands for the euclidian norm of the vector (v₁, . . . , v_J).

Assumption iii) implies in particular that conditionally on G, the functions of M are integrable with respect top_J and that forf ∈ M :

E_G(f(ω, V)) = Z

R^J

f^J×p_J(ω, v₁, ..., v_J)dv₁...dv_J.

The divergence operator LetU = (U_i)_i∈N∗ ∈ P with U_i ∈ S.We define δ:P → S by δ_i(U) : =−(∂_vi(π_iU_i) +U_i1_{pJ>0}∂_i^πlnp_J),

δ(U) = XJ

i=1

δ_i(U) ForF ∈ S, let L(F) =δ(DF).

2.2 Duality and integration by parts formulae

In our framework the duality between δ and D is given by the following proposition.

Proposition 1 Assume H0 andH1, then ∀F ∈ S and ∀U ∈ P we have

E_G(hDF, UiJ) =E_G(F δ(U)). (4)

Proof: By definition, we haveE_G(hDF, UiJ) =PJ

i=1E_G(D_iF ×U_i) and fromH1 E_G(D_iF×U_i) =

Z

R^J

∂_vi(f^J)π_i u^J_i p_J(ω, v₁, ..., v_J)dv₁...dv_J recalling that {πi >0} ⊂Oi, we obtain from Fubini’s theorem

E_G(D_iF×U_i) = Z

RJ−1

Z _bi

ai

∂_vi(f^J)π_i u^J_i p_J(ω, v₁, ..., v_J)dv_i

dv₁..dv_i−1dv_i+1...dv_J. By using the classical integration by parts formula, we have

Z bi

ai

∂_vi(f^J)π_i u^J_i p_J(ω, v₁, ..., v_J)dv_i = [f^Jπ_iu^J_ip_J]^b_aⁱ_i− Z bi

ai

f^J∂_vi(u^J_iπ_ip_J)dv_i.

Now if −∞< a_i < b_i <+∞, we have π_i(a_i) = 0 =π_i(b_i) and [f^Jπ_iu^J_ip_J]^b_aⁱ_i = 0. Moreover since f^J, u^J_i and πi belong to M, we deduce from H1 iii) that lim_|vi|→+∞(f^Jπiu^J_ipJ) = 0 and we obtain that for all a_i,b_i such that−∞ ≤a_i < b_i ≤+∞:

Z bi

ai

∂_vi(f^J)π_i u^J_i p_J(ω, v₁, ..., v_J)dv_i =− Z bi

ai

f^J∂_vi(u^J_iπ_ip_J)dv_i,

Observing that∂_vi(u^J_iπ_ip_J) = (∂_vi(u^J_iπ_i) +u^J_i1_{pJ>0}∂_i^π(lnp_J))p_J, the proposition is proved.

⋄ We have the following straightforward computation rules.

Lemma 1 Let φ:R^d→R be a smooth function and F = (F¹, ..., F^d)∈ S^d. Thenφ(F)∈ S and Dφ(F) =

Xd

r=1

∂_rφ(F)DF^r. (5)

If F ∈ S and U ∈ P then

δ(F U) =F δ(U)− hDF, UiJ. (6)

Moreover, for F = (F¹, ..., F^d)∈ S^d, we have Lφ(F) =

Xd

r=1

∂_rφ(F)LF^r− Xd

r,r^′=1

∂_r,r^′φ(F)D

DF^r, DF^r′E

J. (7)

The first equality is a consequence of the chain rule, the second one follows from the definition of the divergence operatorδ. Combining these equalities (7) follows.

We can now state the main results of this section.

Theorem 1 We assumeH0and H1. LetF = (F¹, ..., F^d)∈ S^d, G∈ S and φ:R^d→R be a smooth bounded function with bounded derivatives. Let Λ∈ G,Λ⊂Λ(F) such that

E(|detγ(F)|^p1_Λ)<∞ ∀p≥1. (8) Then, for every r = 1, ..., d,

EG(∂rφ(F)G) 1Λ=EG(φ(F)Hr(F, G)) 1Λ (9) with

Hr(F, G) = Xd

r^′=1

δ(Gγ^r′,r(F)DF^r′) = Xd

r^′=1

Gδ(γ^r′,r(F)DF^r′)−γ^r′,rD

DF^r′, DGE

J

. (10)

Proof: Using the chain rule DDφ(F), DF^r′E

J =

XJ

j=1

Djφ(F)DjF^r′

= XJ

j=1

( Xd

r=1

∂_rφ(F)D_jF^r)D_jF^r′ = Xd

r=1

∂_rφ(F)σ^r,r′(F)

so that ∂_rφ(F)1_Λ = 1_ΛPd r^′=1

DDφ(F), DF^r′E

Jγ^r′,r(F). Since F ∈ S^d it follows that φ(F) ∈ S and σ^r,r′(F)∈ S.Moreover, since detγ(F)1_Λ∈ ∩p≥1L^pit follows thatγ^r,r′(F)1_Λ∈ S.SoGγ^r′,r(F)DF^r′1_Λ∈

P and the duality formula gives:

E_G(∂_rφ(F)G) 1_Λ = Xd

r^′=1

E_GD

Dφ(F), Gγ^r′,r(F)DF^r′E

J

1_Λ

= Xd

r^′=1

E_G

φ(F)δ(Gγ^r′,r(F)DF^r′) 1_Λ.

⋄

We can extend this integration by parts formula.

Theorem 2 Under the assumptions of Theorem 1, we have for every multi-indexβ = (β₁, . . . , β_q)∈ {1, . . . , d}^q

E_G(∂_βφ(F)G) 1_Λ=E_G

φ(F)H_β^q(F, G)

1_Λ (11)

where the weights H^q are defined recursively by (10) and H_β^q(F, G) =H_β1

F, H_(β^q−1

2,...,βq)(F, G)

. (12)

Proof: The proof is straightforward by induction. For q = 1, this is just Theorem 1. Now assume that Theorem 2 is true forq ≥1 and let us prove it forq+ 1. Letβ = (β₁, . . . , β_q+1)∈ {1, . . . , d}^q+1, we have

EG(∂_βφ(F)G) 1Λ=EG ∂_{(β2,...,βq+1)}(∂_β1φ(F))G

1Λ=EG

∂_β1φ(F)H_(β^q_2,...,β

q+1)(F, G) 1Λ

and the result follows. ⋄

2.3 Estimations of H^q

2.3.1 Iterated derivative operators, Sobolev norms

In order to estimate the weights H^q appearing in the integration by parts formulae of the previous section, we need first to define iterations of the derivative operator. Let α = (α₁, . . . , α_k) be a multi-index, with α_i ∈ {1, . . . , J}, for i = 1, . . . , k and |α| = k. For F ∈ S, we define recursively D_(α^k

1,...,αk)F =D_αk(D^k−1_(α

1,...,αk−1)F) and D^kF =

D_(α^k_1,...,α

k)F

αi∈{1,...,J}.

Remark thatD^kF ∈R^J⊗k and consequently we define the norm ofD^kF as

|D^kF|= vu ut

XJ

α1,...,αk=1

|D_(α^k

1,...,αk)F|². Moreover, we introduce the following norms, for F ∈ S:

|F|1,l = Xl

k=1

|D^kF|, |F|l=|F|+|F|1,l= Xl

k=0

|D^kF|. (13)

ForF = (F¹, . . . , F^d)∈ S^d:

|F|1,l = Xd

r=1

|F^r|1,l, |F|l= Xd

r=1

|F^r|l, and similarly for F = (F^r,r′)_r,r^′_=1,...,d

|F|1,l = Xd

r,r^′=1

|F^r,r′|1,l, |F|l= Xd

r,r^′=1

|F^r,r′|l.

Finally for U = (Ui)i≤J ∈ P, we have D^kU = (D^kUi)i≤J and we define the norm ofD^kU as

|D^kU|= vu utX^J

i=1

|D^kU_i|². We can remark that for k= 0, this gives |U|=p

hU, UiJ. Similarly to (13), we set

|U|1,l = Xl

k=1

|D^kU|, |U|l=|U|+|U|1,l = Xl

k=0

|D^kU|.

Observe that forF, G∈ S, we haveD(F×G) =DF×G+F×DG. This leads to the following useful inequalities

Lemma 2 Let F, G∈ S and U, V ∈ P, we have

|F ×G|l≤2^l X

l1+l2≤l

|F|l1|G|l2, (14)

| hU, ViJ|l≤2^l X

l1+l2≤l

|U|l1|V|l2. (15) We can remark that the first inequality is sharper than the following one |F ×G|l ≤ C_l|F|l|G|l. Moreover from (15) withU =DF and V =DG (F, G,∈ S) we deduce

| hDF, DGiJ|l≤2^l X

l1+l2≤l

|F|1,l1+1|G|1,l2+1 (16)

and as an immediate consequence of (14) and (16), we have for F, G, H ∈ S:

|HhDF, DGiJ|l≤2^2l X

l1+l2+l3≤l

|F|1,l1+1|G|1,l2+1|H|l3. (17)

Proof: We just prove (15), since (14) can be proved on the same way. We first give a bound for D^khU, ViJ = (D_α^khU, ViJ)α∈{1,...,J}^k. For a multi-indexα= (α₁, ..., α_k), withα_i ∈ {1, . . . , J}, we note α(Γ) = (α_i)_i∈Γ, where Γ⊂ {1, . . . , k} and α(Γ^c) = (α_i)_{i /∈Γ}. We have

D^k_αhU, ViJ = XJ

i=1

D_α^k(U_iV_i) = Xk

k^′=0

X

|Γ|=k^′

XJ

i=1

D^k_α(Γ)^′ U_i×D_α(Γ^k−kc^′)V_i.

LetW^i,Γ= (W_α^i,Γ)α∈{1,...,J}^k = (D_α(Γ)^k′ U_i×D^k−k_α(Γc^′)V_i)α∈{1,...,J}^k, we have the equality inR^J⊗k : D^khU, ViJ =

Xk

k^′=0

X

|Γ|=k^′

XJ

i=1

W^i,Γ. This gives

|D^khU, ViJ| ≤ Xk

k^′=0

X

|Γ|=k^′

| XJ

i=1

W^i,Γ|, where

| XJ

i=1

W^i,Γ|= vu ut

XJ

α1,...,αk=1

| XJ

i=1

Wα^i,Γ|². But from Cauchy Schwarz inequality, we have

| XJ

i=1

W_α^i,Γ|²=| XJ

i=1

D^k_α(Γ)^′ U_i×D_α(Γ^k−kc^′)V_i|² ≤ XJ

i=1

|D_α(Γ)^k′ U_i|²× XJ

i=1

|D_α(Γ^k−kc^′)V_i|². Consequently we obtain

| XJ

i=1

W^i,Γ| ≤ vu ut

XJ

α1,...,αk=1

XJ

i=1

|D^k_α(Γ)^′ U_i|²× XJ

i=1

|D_α(Γ^k−kc^′)V_i|²

= |D^k′U| × |D^k−k′V|.

This last equality results from the fact that we sum on different index sets ( Γ and Γ^c). This gives

D^khU, ViJ

≤

Xk

k^′=0

X

|Γ|=k^′

D^k′U

D^k−k′V =

Xk

k^′=0

C_k^k′ D^k′U

D^k−k′V

≤ Xk

k^′=0

C_k^k′|U|k^′|V|k−k^′ ≤2^k( X

l1+l2=k

|U|l1|V|l2).

Summing on k= 0, ..., l we deduce (16).

⋄

2.3.2 Estimation of |γ(F)|l

We give in this section an estimation of the derivatives ofγ(F) in terms of detσ(F) and the derivatives of F. We assume that ω ∈Λ(F).

In what follows C_l,d is a constant depending eventually on the order of derivation l and the dimensiond.

Proposition 2 Let F ∈ S^d, we have ∀l∈N

|γ(F)|l ≤ C_l,d X

l1+l2≤l

|F|^2(d−1)1,l2+1

1

|detσ(F)|+

l1

X

k=1

|F|^2kd1,l1+1

|detσ(F)|^k+1

!

(18)

≤ C_l,d 1

|detσ(F)|^l+1(1 +|F|^2d(l+1)1,l+1 ). (19) Before proving Proposition 2, we establish a preliminary lemma.

Lemma 3 for every G∈S, G >0 we have

1 G

l

≤C_l





 1 G+

Xl

k=1

1 G^k+1

X

k≤r1+...+rk≤l r1,...,rk≥1

Yk

i=1

|D^riG|





≤C_l(1 G+

Xl

k=1

1

G^k+1 |G|^k1,l). (20)

Proof: ForF ∈ S^d andφ:R^d→R aC^∞ function, we have from the chain rule D_(α^k_1,...,α

k)φ(F) = Xk

|β|=1

∂_βφ(F) X

Γ1∪...∪Γ|β|={1,...,k}



 Y|β|

i=1

D^|Γ_α(Γ^i|

i)F^βi



, (21)

where β ∈ {1, . . . , d}^|β| and P

Γ1∪...∪Γ|β| denotes the sum over all partitions of{1, . . . , k} with length

|β|. In particular, for G∈ S,G >0 and for φ(x) = 1/x, we obtain

|D_α^k(1

G)| ≤C_k Xk

k^′=1

1 G^k′+1

X

Γ1∪...∪Γ_k′={1,...,k}

k^′

Y

i=1

|D_α(Γ^|Γi|

i)G|

!

. (22)

We deduce then that

|D^k(1

G)| ≤ C_k Xk

k^′=1

1 G^k′+1

X

Γ1∪...∪Γk′={1,...,k}

k^′

Y

i=1

D_α(Γ^|Γi|

i)G RJ⊗k

,

= C_k Xk

k^′=1

1 G^k′+1

X

Γ1∪...∪Γ_k′={1,...,k}

k^′

Y

i=1

|D^|Γi|G|

! ,

= C_k Xk

k^′=1

1 G^k′+1

X

r1+...+r_k′=k r1,...,r_k′≥1

k^′

Y

i=1

|D^riG|

! ,

and the first part of (20) is proved. The proof of the second part is straightforward. ⋄ With this lemma, we can prove Proposition 2.

Proof: Proposition 2. We have on Λ(F)

γ^r,r′(F) = 1

detσ(F)σˆ^r,r′(F),

where ˆσ(F) is the algebraic complement of σ(F). But recalling that σ^r,r′(F) = D

D^rF, D^r′FE

J we have

|detσ(F)|l≤C_l,d|F|^2d1,l+1 and |σ(F)ˆ |l ≤C_l,d|F|^2(d−1)_1,l+1 . (23) Applying inequality (14), this gives

|γ(F)|l≤C_l,d X

l1+l2≤l

|(detσ(F))⁻¹|l1|σ(F)ˆ |l2. From Lemma 3 and (23), we have

|(detσ(F))⁻¹|l1 ≤C_l1

1

|detσ(F)|+

l1

X

k=1

|F|^2kd1,l1+1

|detσ(F)|^k+1

! .

Putting together these inequalities, we obtain the inequality (18) and consequently (19). ⋄

2.3.3 Some bounds on H^q

Now our goal is to establish some estimates for the weights H^q in terms of the derivatives of G, F, LF and γ(F).

Theorem 3 ForF ∈ S^d , G∈ S and for all q∈N^∗ there exists an universal constant C_q,d such that for every multi-index β= (β₁, .., β_q)

H_β^q(F, G)

≤ C_q,d|G|_q(1 +|F|_q+1)^(6d+1)q

|detσ(F)|^3q−1 (1 + Xq

j=1

X

k1+...+kj≤q−j

Yj

i=1

|L(F)|ki),

≤ C_q,d|G|q(1 +|F|q+1)^(6d+1)q

|detσ(F)|^3q−1 (1 +|LF|^qq−1).

Proof: ForF ∈ S^d, we define the linear operator Tr:S → S, r= 1, ..., d by T_r(G) =hDG,(γ(F)DF)^ri,

where (γ(F)DF)^r =Pd

r^′=1γ^r′,r(F)DF^r′. Notice that

T_r(G×G^′) =GT_r(G^′) +G^′T_r(G). (24) Moreover, for a multi-index β = (β₁, .., β_q) we define by induction T_β(G) =T_βq(T_{(β1,...,βq−1)}(G)). We also make the convention that if β is the void multi-index, then T_β(G) = G. Finally we denote by L^γr(F) =Pd

r^′=1δ(γ^r′,r(F)DF^r′).With this notation we have H_r(F, G) = GL^γ_r(F)−T_r(G), H_β^q(F, G) = H_β1(F, H_(β^q−1

2,...,βq)(F, G)).

We will now give an explicite expression of H_β^q(F, G). In order to do this we have to introduce some more notation. Let Λ_j ={λ₁, . . . , λ_j} ⊂ {1, . . . , q}such that |Λ_j|=j. We denote byP(Λ_j) the set of the partitions Γ = (Γ₀,Γ₁, ...,Γ_j) of{1, ..., q} \Λ_j . Notice that we accept that Γ_i, i= 0,1, ..., j may be void sets. Moreover, for a multi-index β = (β₁, .., β_q) we denote by Γ_i(β) = (β_k¹

i, ..., β_k^p

i) where Γ_i ={k¹_i, ..., k^p_i}.With this notation we can prove by induction and using (24) that

H_β^q(F, G) =T_β(G) + Xq

j=1

X

Λj⊂{1,...q}

X

Γ∈P(Λj)

c_β,ΓT_Γ0(β)(G) Yj

i=1

T_Γi(β)(L^γ_β

λi(F)) (25)

wherec_β,Γ∈ {−1,0,1}.

We first give an estimation of|T_β(G)|l, forl ≥0 andβ = (β₁, . . . , β_q). We proceed by induction.

Forq = 1 and 1≤r≤d, we have

|T_r(G)|l=| hDG,(γ(F)DF)^ri |l

and using (17) we obtain

|T_r(G)|l ≤C_l X

l1+l2+l3≤l

|γ(F)|l1|G|l2+1|F|l3+1 ≤ |G|l+1|F|l+1

Xl

l1=0

|γ(F)|l1,

where C_l is a constant which depends on l only. We obtain then by induction for every multi-index β = (β₁, . . . , β_q)

|T_β(G)|_l≤C_l,q|G|l+q|F|^ql+q

X

l1+...+lq≤l+q−1

Yq

i=1

|γ(F)|li. (26) In particular this gives forl= 0

|T_β(G)| ≤C_q|G|q|F|^qqP_q(γ(F)),

with

P_q(γ(F)) = X

l1+...+lq≤q−1

Yq

i=1

|γ(F)|li, q≥1.

To complete the notation, we note P₀(γ(F)) = 1. We obtain

T_Γi(β)(L^γ_β

λi(F))

≤ C_q L^γ_β

λi(F)

|Γi(β)||F|^|Γ_|Γⁱ_i^(β)|_(β)|P_|Γi(β)|(γ(F)).

We turn now to the estimation of|L^γ_r(F)|l. From the properties of the divergence operator δ (see Lemma 1)

δ(γ(F)DF) =γ(F)δ(DF)− hDγ(F), DFiJ. It follows from (14) and (16) that

|L^γ_r(F)|l ≤C_l|γ(F)|l+1(|δ(DF)|l+|F|l+1)≤C_l|γ(F)|l+1(1 +|LF|l)(1 +|F|l+1), and we get

T_Γi(β)(L^γ_β

λi(F))

≤C_q|γ(F)||Γi(β)|+1(1 +|LF||Γi(β)|)(1 +|F||Γi(β)|+1)|F|^|Γ_|Γⁱ_i^(β)|_(β)|P_|Γi(β)|(γ(F)). (27) Reporting these inequalities in (25) and recalling that|Γ0(β)|+. . .+|Γj(β)|=q−j we deduce :

|H_β^q(F, G)| ≤ |T_β(G)|+C_q,d Xq

j=1

X

k0+···+kj=q−j

|G|k0|F|^k_k⁰0P_k0(γ(F)) Yj

i=1

|γ(F)|ki+1P_ki(γ(F))

|F|^k_kⁱ_i(1 +|F|ki+1)(1 +|LF|ki)

(28) Now, forq ≥1, we have from (19) :

P_q(γ(F))≤C_q 1

|detσ(F)|^2q−1(1 +|F|q)^4dq, so the following inequality holds forq ≥0 :

P_q(γ(F))≤C_q 1

|detσ(F)|^2q(1 +|F|q)^4dq. We obtain then fork₀, k₁, . . . , k_j ∈N such thatk₀+. . .+k_j =q−j

Yj

i=0

P_ki(γ(F))≤C_q 1

|detσ(F)|^2(q−j)(1 +|F|q−j)^4d(q−j) (29) and once again from (19)

Yj

i=1

|γ(F)|ki+1≤C_q 1

|detσ(F)|^q+j(1 +|F|q−j+2)^2d(q+j) (30)

it yields finally Yj

i=0

P_ki(γ(F)) Yj

i=1

|γ(F)|ki+1 ≤Cq 1

|detσ(F)|^3q−j(1 +|F|^q−j+2)^6dq−2dj. Turning back to (28), it follows that

H_β^q(F, G)

≤ C_q,d|G|q(1 +|F|q+1)^(6d+1)q

|detσ(F)|^3q−1 (1 + Xq

j=1

X

k1+...+kj≤q−j

Yj

i=1

|L(F)|ki),

and Theorem 3 is proved. ⋄

3 Stochastic equations with jumps

3.1 Notations and hypotheses

We consider a Poisson point process p with state space (E, B(E)),where E =R^d×R₊. We refer to [I.W] for the notation. We denote byN the counting measure associated top, we haveN([0, t)×A) =

#{0 ≤s < t;p_s ∈A} for t≥0 and A ∈B(E). We assume that the associated intensity measure is given byNb(dt, dz, du) =dt×dµ(z)×1_[0,∞)(u)du where (z, u)∈E =R^d×R₊ and µ(dz) =h(z)dz.

We are interested in the solution of theddimensional stochastic equation X_t=x+

Z t

0

Z

E

c(z, X_s−)1{u<γ(z,Xs−)}N(ds, dz, du) + Z t

0

g(X_s)ds. (31) We remark that the infinitesimal generator of the Markov processX_t is given by

Lψ(x) =g(x)∇ψ(x) + Z

R^d

(ψ(x+c(z, x))−ψ(x))K(x, dz)

where K(x, dz) =γ(z, x)h(z)dz depends on the variable x ∈R^d.See [F.1] for the proof of existence and uniqueness of the solution of the above equation.

Our aim is to give sufficient conditions in order to prove that the law ofX_tis absolutely continuous with respect to the Lebesgue measure and has a smooth density. In this section we make the following hypotheses on the functions γ, g, h and c.

Hypothesis 3.0We assume that γ, g, h and c are infinitely differentiable functions in both vari- ables zand x. Moreover we assume thatg and its derivatives are bounded and that lnh has bounded derivatives

Hypothesis 3.1. We assume that there exist two functionsγ, γ :R^d→R₊ such that C≥γ(z)≥γ(z, x)≥γ(z)≥0, ∀x∈R^d

whereC is a constant.

Hypothesis 3.2. i)We assume that there exists a non negative and bounded functionc:R^d→R₊ such thatR

R^dc(z)dµ(z)<∞ and

|c(z, x)|+

∂_z^β∂_x^αc(z, x)

≤c(z) ∀z, x∈R^d. We need this hypothesis in order to estimate the Sobolev norms.

ii)There exists a measurable functionbc:R^d→R₊ such thatR

R^dbc(z)dµ(z)<∞ and ∇xc×(I+∇xc)⁻¹(z, x)

≤bc(z), ∀(z, x)∈R^d×R^d. In order to simplify the notations we assume thatbc(z) =c(z).

iii) There exists a non negative functionc:R^d→R₊ such that for every z∈R^d Xd

r=1

h∂_zrc(z, x), ξi² ≥c²(z)|ξ|², ∀ξ ∈R^d and we assume that there exists θ >0 such that

lim_a→+∞ 1 lna

Z

{c²≥1/a}

γ(z)dµ(z) =θ.

Remark : assumptions ii) and iii) give sufficient conditions to prove the non degeneracy of the Malliavin covariance matrix as defined in the previous section. In particular the second part of iii) implies that c² is a (p, t) broad function (see [B.G.J.]) forp/t < θ. Notice that we may havec(z) = 0 for somez∈R^d.

We add to these hypotheses some assumptions on the derivatives ofγ and lnγ with respect tox and z. Forl≥1 we use the notation :

γ^x,l(z) = sup

x

sup

1≤|β|≤l|∂_β,xγ(z, x)|, γ^x,l_ln(z) = sup

x

sup

1≤|β|≤l|∂_β,xlnγ(z, x)|, γ^z,l_ln(z) = sup

x sup

1≤|β|≤l|∂_β,zlnγ(z, x)|.

Hypothesis 3.3. We assume that lnγ has bounded derivatives with respect to z (that is γ^z,l_ln(z) is bounded) and that γ has bounded derivatives with respect to x such that ∀z ∈ R^d, γ^x,l(z) ≤ γ^x,l; moreover we assume that

sup

z^∗∈Rd

Z

B(z^∗,1)

γ(z)dµ(z)<+∞.

We complete this hypothesis with two alternative hypotheses.

a) (weak dependence onx) We assume that∀l≥1 Z

R^d

γ^x,l_ln(z)γ(z)dµ(z)<∞.

b) (strong dependence onx) We assume that lnγ has bounded derivatives with respect to xsuch that∀l≥1

∀z∈R^d, γ^x,l_ln(z)≤γ^x,l_ln.

Remark : if µ is the Lebesgue measure ( case h = 1) and if γ does not depend on z then γ^x,l_ln is constant and consequently hypothesis 3.3.afails. Conversely, ifγ(z, x) =γ(z) then hypothesis 3.3.ais satisfied as soon as lnγ has bounded derivatives. This last case corresponds to the standard case where the law of the amplitude of the jumps does not depend on the position ofX_t. Under Hypothesis 3.3.a we are in a classical situation where the divergence does not blow up and this leads to an integration by part formula with bounded weights (see Proposition 4 and Lemma 11). On the contrary under assumption 3.3.b, the divergence can blow up as well as the weights appearing in the integration by part formula.

3.2 Main results and examples

Our methodology to study the regularity of the law of the random variableX_tis based on the following result. Let ˆp_X(ξ) = E(e^ihξ,Xi) be the Fourier transform of a d-dimensional random variable X then using the Fourier inversion formula, one can prove that if R

R^d|ξ|^p|pˆ_X(ξ)|dξ < ∞ for p > 0 then the law ofX is absolutely continuous with respect to the Lebesgue measure on R^dand its density is C^[p], where [p] denotes the entire part of p.

To apply this result, we just have to bound the Fourier transform ofX_t in terms of 1/|ξ|.This is done in the next proposition. The proof of this proposition needs a lot of steps that we detail in the next sections and it will be given later.

Proposition 3 Let B_M ={z ∈ R^d;|z|< M}, then under hypotheses 3.0., 3.1. 3.2. and 3.3 we have for all M ≥1, for q ≥1 and t >0 such that4d(3q−1)/t < θ

a) if3.3.a holds

|pˆXt(ξ)| ≤ t Z

B^c_M−1

c²(z)γ(z)dµ(z)1

2|ξ|²+|ξ|te^Ct Z

B_M^c

c(z)γ(z)dµ(z) + C_q

|ξ|^q. b) if3.3.b holds

|pˆXt(ξ)| ≤ t Z

B_M−1^c

c²(z)γ(z)dµ(z)1

2|ξ|²+|ξ|te^Ct Z

B_M^c

c(z)γ(z)dµ(z) + C_q(1 +µ(B_M+1)^q)

|ξ|^q . We can remark that if θ= +∞ then the result holds∀q ≥1 and∀t >0.

By choosing M judiciously as a function ofξ in the inequalities given in Proposition 3, we obtain

|pˆ_Xt(ξ)| ≤ C/|ξ|^p for some p > 0 and this permits us to deduce some regularity for the density of Xt. The next theorem precise the optimal choice ofM with respect to ξ and permits us to derive the regularity of the law of the process X_t.

Theorem 4 We assume that hypotheses 3.0., 3.1., 3.2 and3.3. hold.

a) Assuming 3.3.a, the law of X_t admits a density C^k if t >(3k+ 3d−1)^4d_θ. In the case θ=∞, the law of X_t admits a density C^∞.

b)Assuming 3.3.b and the two following hypotheses A1: ∃p₁, p₂ >0 such that :

lim sup

M

M^p1 Z

B^c_M

c(z)γ(z)dµ(z) <+∞; lim sup

M

M^p2 Z

B^c_M

c²(z)γ(z)dµ(z)<+∞; A2: ∃ρ >0 such thatµ(B_M)≤CM^ρ where B_M ={z∈R^d;|z|< M};

case 1: ifθ= +∞ then the law ofX_tadmits a density C^k withk <min(p₁/ρ−1−d, p₂/ρ−2−d) if min(p1/ρ−1−d, p2/ρ−2−d)≥1 .

case 2 : if 0 < θ < ∞ let q^∗(t, θ) = [¹₃(_4d^tθ + 1)]; then the law of X_t admits a density C^k for k < sup0<r<1/ρmin(rp₁ −1 −d, rp₂ −2−d, q^∗(t, θ)(1 −rρ)−d), if for some 0 < r < 1/ρ, min(rp₁−1−d, rp₂−2−d, q^∗(t, θ)(1−rρ)−d)≥1.

Proof:

a) Assuming 3.3.a and letting M go to infinity in the right-hand side of the inequality given in Proposition 3 , we deduce

|pˆ_Xt(ξ)| ≤C/|ξ|^q, and the result follows.

b) FromA1, forM large enough, we have Z

B^c_M

c(z)γ(z)dµ(z)≤C/M^p1