Integration by parts formula with respect to jump times for stochastic differential equations

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Integration by parts formula with respect to jump times for stochastic differential equations

Vlad Bally, Emmanuelle Clement

To cite this version:

Vlad Bally, Emmanuelle Clement. Integration by parts formula with respect to jump times for stochas- tic differential equations. Stochastic Analysis 2010, Springer-Verlag, pp.7-29, 2010. �hal-00472657�

Integration by parts formula with respect to jump times for stochastic differential equations

Vlad Bally^∗† Emmanuelle Cl´ement^∗ April, 7 2010

Abstract

We establish an integration by parts formula based on jump times in an abstract framework in order to study the regularity of the law for processes solution of stochastic differential equations with jumps.

2010 MSC. Primary: 60H07, Secondary: 60G55, 60G57

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

1 Introduction

We consider the one dimensional equation X_t=x+

Z _t

0

c(u, a, X_u−)dN(u, a) + Z _t

0

g(u, X_u)du (1)

where N is a Poisson point measure of intensity measure µ on some abstract measurable space E.

We assume thatcand gare infinitely differentiable with respect totand x, have bounded derivatives of any order and have linear growth with respect tox. Moreover we assume that the derivatives of c are bounded by a function c such that R

Ec(a)dµ(a) < ∞. Under these hypotheses the equation has a unique solution and the stochastic integral with respect to the Poisson point measure is a Stieltjes integral.

∗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.

†Acknowledgement : Part of this work has been done during a visit of the first author to the Institute Mittag-Leffler (Djursholm, Sweden)

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. If E = R^m and if the measure µ admits a smooth density h then one may develop a Malliavin calculus based on the amplitudes of the jumps in order to solve this problem. This has been done first in [4] and then in [3]. But if µ is a singular measure this approach fails and one has to use the noise given by the jump times of the Poisson point measure in order to settle a differential calculus analogous to the Malliavin calculus.

This is a much more delicate problem and several approaches have been proposed. A first step is to prove that the law of X_t is absolutely continuous with respect to the Lebesgue measure, without taking care of the regularity. A first result in this sense was obtained by Carlen and Pardoux in [5]

and was followed by a lot of other papers (see [6], [7], [11], [13]). The second step is to obtain the regularity of the density. Recently two results of this type have been obtained by Ishikawa and Kunita in [10] and by Kulik in [12]. In both cases one deals with an equation of the form

dX_t=g(t, X_t)dt+f(t, X_t−)dU_t (2) whereU is a L´evy process. The above equation is multi-dimensional (let us mention that the method presented in our paper may be used in the multi-dimensional case as well, but then some technical problems related to the control of the Malliavin covariance matrix have to be solved - and for simplicity we preferred to leave out this kind of difficulties in this paper). Ishikawa and Kunita in [10] used the finite difference approach given by J. Picard in [14] in order to obtain sufficient conditions for the regularity of the density of the solution of an equation of type (1) (in a somehow more particular form, closed to linear equations). The result in that paper produces a large class of examples in which we get a smooth density even for an intensity measure which is singular with respect to the Lebesgue measure. The second approach is due to Kulik [12]. He settled a Malliavin type calculus based on perturbations of the time structure in order to give sufficient conditions for the smoothness of the density. In his paper the coefficientf is equal to one so the non degeneracy comes from the drift term g only. As before, he obtains the regularity of the density even if the intensity measure µis singular.

He also proves that under some appropriate conditions, the density is not smooth for a smalltso that one has to wait before the regularization effect of the noise produces a regular density.

The result proved in our paper is the following. We consider the function α(t, a, x) =g(x)−g(x+c(t, a, x)) + (g∂_xc+∂_tc)(t, a, x).

Except the regularity and boundedness conditions ongandcwe consider the following non degeneracy

assumption. There exists a measurable functionα such that|α(t, a, x)| ≥α(a)>0 for every (t, a, x)∈ R₊×E×R.We assume that there exists a sequence of subsetsE_n↑E such thatµ(E_n)<∞ and

lim_n→∞ 1 µ(E_n)ln(

Z

En

1

α(a)dµ(a)) =θ <∞.

Ifθ= 0 then, for every t >0,the law ofX_t has a C^∞density with respect to the Lebesgue measure.

Suppose now that θ >0 and let q ∈ N.Then, for t >16θ(q+ 2)(q+ 1)² the law ofX_t has a density of class C^q.Notice that for small twe are not able to prove that a density exists and we have to wait for a sufficiently large tin order to obtain a regularization effect.

In the paper of Kulik [12] one takes c(t, a, x) = a so α(t, a, x) = g(x)−g(x+c(t, a, x)). Then the non degeneracy condition concerns just the drift coefficient g.And in the paper of Ishikawa and Kunita the basic example (which corresponds to the geometric L´evy process) isc(t, a, x) =xa(e^a−1) and g constant. So α(t, a, x) =a(e^a−1) ∼a² as a→0. The drift coefficient does not contribute to the non degeneracy condition (which is analogous to the uniform ellipticity condition).

The paper is organized as follows. In Section 2 we give an integration by parts formula of Malliavin type. This is analogous to the integration by parts formulas given in [2] and [1]. But there are two specific points: first of all the integration by parts formula take into account the border terms (in the above mentioned papers the border terms cancel because one makes use of some weights which are null on the border; but in the paper of Kulik [12] such border terms appear as well). The second point is that we use here a ”one shot” integration by parts formula: in the classical gaussian Malliavin calculus one employs all the noise which is available - so one derives an infinite dimensional differential calculus based on ”all the increments” of the Brownian motion. The analogous approach in the case of Poisson point measures is to use all the noise which comes from the random structure (jumps). And this is the point of view of almost all the papers on this topic. But in our paper we use just ”one jump time” which is chosen in a cleaver way (according to the non degeneracy condition). In Section 3 we apply the general integration by parts formula to stochastic equations with jumps. The basic noise is given by the jump times.

2 Integration by parts formula

2.1 Notations-derivative operators

The abstract framework is quite similar to the one developed in [2] but we introduce here some modifications in order to take into account the border terms appearing in the integration by parts

formula. 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 variableJ,G measurable, with values in N. Our aim is to establish a differential calculus based on the variables (V_i), conditionally onG. In order to derive an integration by parts formula, we need some assumptions on the random variables (V_i). The main hypothesis is that conditionally on G, the law of V_i admits a locally smooth density with respect to the Lebesgue measure.

H0. i) Conditionally on G, the random variables (Vi)_1≤i≤J are independent and for each i ∈ {1, . . . , J} the law of V_i is absolutely continuous with respect to the Lebesgue measure. We note p_i the conditional density.

ii) For all i ∈ {1, . . . , J}, there exist some G measurable random variables a_i and b_i such that −∞< a_i < b_i <+∞, (ai, b_i)⊂ {pi >0}. We also assume that p_i admits a continuous bounded derivative on (ai, bi) and that lnpi is bounded on (ai, bi).

We define now the class of functions on which this differential calculus will apply. We consider in this paper functionsf : Ω×R^N∗ →Rwhich can be written as

f(ω, v) =

∞

X

m=1

f^m(ω, v₁, ..., v_m)1_{J(ω)=m} (3) wheref^m : Ω×R^m→RareG × B(R^m)−measurable functions.

In the following, we fix L ∈ N and we will perform integration by parts L times. But we will use another set of variables for each integration by parts. So for 1 ≤ l ≤ L, we fix a set of indices I_l ⊂ {1, . . . , J} such that if l 6= l^′, I_l∩I_l^′ = ∅. In order to do l integration by parts, we will use successively the variables V_i, i ∈I_l then the variables V_i, i∈ I_l−1 and end with V_i, i ∈I₁. Moreover, given lwe fix a partition (Λ_l,i)_i∈Il of Ω such that the sets Λ_l,i∈ G, i∈I_l. Ifω ∈Λ_l,i, we will use only the variable V_i in our integration by parts.

With these notations, we define our basic spaces. We consider in this paper random variablesF = f(ω, V) whereV = (V_i)_iandf is given by (3). To simplify the notation we writeF =f^J(ω, V₁, . . . , V_J) so that conditionally on G we haveJ =m andF =f^m(ω, V₁, . . . , V_m). We denote byS⁰ the space of random variables F =f^J(ω, V₁, . . . , V_J) where f^J is a continuous function onO_J =QJ

i=1(a_i, b_i) such that there exists aG measurable random variable C satisfying

sup

v∈OJ

|f^J(ω, v)| ≤C(ω)<+∞ a.e. (4)

We also assume that f^J has left limits (respectively right limits) in a_i (respectively inb_i). Let us be more precise.

With the notations V_(i) = (V₁, ..., V_i−1, V_i+1, ..., V_J) and (V_(i), v_i) = (V₁, . . . , V_i−1, v_i, V_i+1, . . . , V_J) forv_i ∈(a_i, b_i) our assumption is that the following limits exist and are finite:

ε→0limf^J(ω, V_(i), a_i+ε) :=F(a⁺_i ), lim

ε→0f^J(ω, V_(i), b_i−ε) :=F(b⁻_i ). (5) Now fork≥1,S^k(I_l) denotes the space of random variablesF =f^J(ω, V1, . . . , VJ)∈ S⁰, such that f^J admits partial derivatives up to order k with respect to the variables v_i, i ∈ I_l and these partial derivatives belong to S⁰.

We are now able to define our differential operators.

The derivative operators. We define D_l :S¹(I_l)→ S⁰(I_l) : by D_lF := 1_OJ(V)X

i∈Il

1_Λl,i(ω)∂_vif(ω, V),

whereO_J =Q_J

i=1(a_i, b_i).

The divergence operators We note

p_(l) =X

i∈I_l

1_Λl,ip_i, (6)

and we defineδ_l :S¹(I_l)→ S⁰(I_l) by

δ_l(F) =D_lF +F D_llnp_(l) = 1_OJ(V)X

i∈Il

1_Λl,i(∂_viF +F ∂_vilnp_i)

We can easily see that ifF, U ∈ S¹(I_l) we have

δ_l(F U) =F δ_l(U) +U D_lF. (7)

The border terms Let U ∈ S⁰(I_l). We define (using the notation (5) ) [U]_l = X

i∈Il

1_Λl,i1_OJ,i(V_(i))((U p_i)(b⁻_i )−(U p_i)(a⁺_i )) withO_J,i=Q

1≤j≤J,j6=i(a_j, b_j)

2.2 Duality and basic integration by parts formula

In our framework the duality between δ_l and D_l is given by the following proposition. In the sequel, we denote by E_G the conditional expectation with respect to the sigma-algebra G.

Proposition 1 Assuming H0 then ∀F, U ∈ S¹(I_l) we have

E_G(U D_lF) =−EG(F δ_l(U)) +E_G[F U]_l. (8) For simplicity, we assume in this proposition that the random variables F and U take value in Rbut such a result can easily be extended toR^d value random variables.

Proof: We haveE_G(U D_lF) =P

i∈Il1_Λl,iE_G1_OJ(V)(∂_vif^J(ω, V)u^J(ω, V)). From H0 we obtain E_G1_OJ(V)(∂_vif^J(ω, V)u^J(ω, V)) =E_G1_OJ,i(V_(i))

Z bi

ai

∂_vi(f^J)u^Jp_i(v_i)dv_i. By using the classical integration by parts formula, we have

Z _bi

ai

∂_vi(f^J)u^Jp_i(v_i)dv_i = [f^Ju^Jp_i]^b_aⁱ_i− Z _bi

ai

f^J∂_vi(u^Jp_i)dv_i. Observing that∂_vi(u^Jp_i) = (∂_vi(u^J) +u^J∂_vi(lnp_i))p_i, we have

E_G(1_OJ(V)∂_vif^Ju^J) =E_G1_OJ,i[(V_(i))f^Ju^Jp_i]^b_aⁱ_i−E_G1_OJ(V)F(∂_vi(U) +U ∂_vi(lnp_i)) and the proposition is proved.

⋄ We can now state a first integration by parts formula.

Proposition 2 Let H0 hold true and letF ∈ S²(I_l),G∈ S¹(I_l)andΦ :R→Rbe a bounded function with bounded derivative. We assume that F =f^J(ω, V) satisfies the condition

mini∈Il

v∈OinfJ

|∂v_if^J(ω, v)| ≥γ(ω), (9)

where γ isG measurable and we define on{γ >0}

(D_lF)⁻¹ = 1_OJ(V)X

i∈Il

1_Λl,i 1

∂vif(ω, V), then

1_{γ>0}E_G(Φ⁽¹⁾(F)G) =−1_{γ>0}E_G(Φ(F)H_l(F, G)) + 1_{γ>0}E_G[Φ(F)G(D_lF)⁻¹]_l (10) with

H_l(F, G) = δ_l(G(D_lF)⁻¹) =Gδ_l((D_lF)⁻¹) +D_lG(D_lF)⁻¹. (11)

Proof: We observe that

D_lΦ(F) = 1_OJ(V)X

i∈Il

1_Λl,i∂_viΦ(F) = 1_OJ(V)Φ⁽¹⁾(F)X

i∈Il

1_Λl,i∂_viF,

so that

D_lΦ(F).D_lF = Φ⁽¹⁾(F)(D_lF)²,

and then 1_{γ>0}Φ⁽¹⁾(F) = 1_{γ>0}D_lΦ(F).(D_lF)⁻¹. Now sinceF ∈ S²(I_l), we deduce that (D_lF)⁻¹ ∈ S¹(I_l) on {γ >0}and applying Proposition 1 with U =G(D_lF)⁻¹ we obtain the result.

⋄

2.3 Iterations of the integration by parts formula

We will iterate the integration by parts formula given in Proposition 2. We recall that if we iterate l times the integration by parts formula, we will integrate by parts successively with respect to the variables (V_i)_i∈Ik for 1 ≤ k ≤ l. In order to give some estimates of the weights appearing in these formulas we introduce the following norm onS^l(∪^l_k=1I_k), for 1≤l≤L.

|F|l =|F|∞+

l

X

k=1

X

1≤l1<...<lk≤l

|Dl1. . . D_lkF|∞, (12) where|.|∞ is defined onS⁰ by

|F|∞= sup

v∈O_J|f^J(ω, v)|.

Forl= 0, we set|F|0 =|F|∞. We remark that we have for 1≤l₁< . . . < l_k ≤l

|D_l1. . . D_lkF|∞= X

i1∈Il1,...,ik∈I_lk

(

k

Y

j=1

1Λ_{lj ,ij})|∂vi1 . . . ∂v_ikF|∞,

and since for each l(Λ_l,i)i∈I_l is a partition of Ω, for ω fixed, the preceding sum has only one term not equal to zero. This family of norms satisfies for F ∈ S^l+1(∪^l+1_k=1I_k) :

|F|_l+1=|D_l+1F|_l+|F|_l so |D_l+1F|_l≤ |F|_l+1. (13) Moreover it is easy to check that ifF, G∈ S^l(∪^l_k=1I_k)

|F G|_l ≤C_l|F|_l|G|_l, (14)

whereC_l is a constant depending on l. Finally for any functionφ∈ C^l(R,R) we have

|φ(F)|_l≤C_l

l

X

k=0

|φ^(k)(F)|_∞|F|^k_l ≤C_l max

0≤k≤l|φ^(k)(F)|_∞(1 +|F|^l_l). (15) With these notations we can iterate the integration by parts formula.

Theorem 1 Let H0 hold true and let Φ :R7→ R a bounded function with bounded derivatives up to order L. Let F =f^J(w, V)∈ S¹(∪^L_l=1I_l) such that

i∈{1,...,J}inf inf

v∈OJ

|∂vif^J(ω, v)| ≥γ(ω), γ ∈[0,1] Gmeasurable (16) then we have for l∈ {1, . . . , L}, G∈ S^l(∪^l_k=1I_k) andF ∈ S^l+1(∪^l_k=1I_k)

1_{γ>0}|EGΦ^(l)(F)G| ≤C_l||Φ||∞1_{γ>0}E_G

|G|_l(1 +|p|0)^lΠ_l(F)

(17) where C_l is a constant depending on l, ||Φ||∞ = sup_x|Φ(x)|, |p|0 = max_l=1,...,L|p_(l)|∞ and Π_l(F) is defined on {γ >0} by

Π_l(F) =

l

Y

k=1

(1 +|(DkF)⁻¹|k−1)(1 +|δk((D_kF)⁻¹)|k−1). (18) Moreover we have the bound

Π_l(F)≤C_l(1 +|lnp|1)^l γ^l(l+2)

l

Y

k=1

(1 +|F|^k−1_k +|DkF|^k−1_k )², (19) where |lnp|1= max_i=1,...,J|(lnp_i)^′|∞.

Proof: We proceed by induction. For l = 1, we have from Proposition 2 since G ∈ S¹(I₁) and F ∈ S²(I₁)

1_{γ>0}E_G(Φ⁽¹⁾(F)G) =−1_{γ>0}E_G(Φ(F)H₁(F, G)) + 1_{γ>0}E_G[Φ(F)G(D₁F)⁻¹]₁. We have on {γ >0}

|H1(F, G)| ≤ |G||δ1((D₁F)⁻¹)|+|D1G||(D1F)⁻¹|,

≤ (|G|∞+|D1G|∞)(1 +|(D1F)⁻¹|∞)(1 +|δ1((D₁F)⁻¹)|∞),

= |G|1(1 +|(D1F)⁻¹|0)(1 +|δ1((D1F)⁻¹)|0).

Turning to the border term [Φ(F)G(D₁F)⁻¹]₁, we check that

|[Φ(F)G(D₁F)⁻¹]₁| ≤ 2||Φ||_∞|G|_∞P

i∈I11_Λ1,i|_∂¹

viF|_∞P

i∈I11_Λ1,i|p_i|_∞,

≤ 2||Φ||∞|G|0|(D1F)⁻¹|0|p|0. This proves the result for l= 1.

Now assume that Theorem 1 is true for l ≥ 1 and let us prove it for l+ 1. By assumption we have G ∈ S^l+1(∪^l+1_k=1I_k) ⊂ S¹(I_l+1) and F ∈ S^l+2(∪^l+1_k=1I_k) ⊂ S²(I_l+1). Consequently we can apply Proposition 2 on I_l+1. This gives

1_{γ>0}E_G(Φ^(l+1)(F)G) =−1_{γ>0}E_G

Φ^(l)(F)H_l+1(F, G)

+ 1_{γ>0}E_G[Φ^(l)(F)G(D_l+1F)⁻¹]_l+1, (20) with

H_l+1(F, G) =Gδ_l+1((D_l+1F)⁻¹) +D_l+1G(D_l+1F)⁻¹, [Φ^(l)(F)G(D_l+1F)⁻¹]_l+1 = X

i∈Il+1

1_Λl+1,i1_OJ,i(V_(i))

(Φ^(l)(F)G 1

∂v_iFp_i)(b⁻_i )−(Φ^(l)(F)G 1

∂v_iFp_i)(a⁺_i )

. We easily see thatH_l+1(F, G)∈ S^l(∪^l_k=1I_k) and so using the induction hypothesis we obtain

1_{γ>0}|EGΦ^(l)(F)H_l+1(F, G)| ≤C_l||Φ||∞1_{γ>0}E_G|Hl+1(F, G)|l(1 +|p|0)^lΠ_l(F), and we just have to bound |Hl+1(F, G)|l on {γ >0}. But using successively (14) and (13)

|Hl+1(F, G)|l ≤ C_l(|G|l|δl+1((D_l+1F)⁻¹)|l+|Dl+1G|l|(Dl+1F)⁻¹)|l,

≤ C_l|G|l+1(1 +|(Dl+1F)⁻¹)|l)(1 +|δl+1((D_l+1F)⁻¹)|l).

This finally gives

|EGΦ^(l)(F)H_l+1(F, G)| ≤C_l||Φ||∞E_G|G|l+1(1 +|p|0)^lΠ_l+1(F). (21) So we just have to prove a similar inequality for E_G[Φ^(l)(F)G(D_l+1F)⁻¹]_l+1. This reduces to consider

E_G X

i∈I_l+1

1_Λl+1,i1_OJ,i(V_(i))(Φ^(l)(F)G 1

∂_viFp_i)(b⁻_i ) = X

i∈I_l+1

1_Λl+1,ip_i(b⁻_i )E_G1_OJ,i(V_(i))(Φ^(l)(F)G 1

∂_viF)(b⁻_i ) (22) since the other term can be treated similarly. Consequently we just have to bound

|EG1_OJ,i(V_(i))(Φ^(l)(F)G 1

∂viF)(b⁻_i )|.

Since all variables satisfy (4), we obtain from Lebesgue Theorem, using the notation (5) E_G1_OJ,i(V_(i))(Φ^(l)(F)G 1

∂viF)(b⁻_i ) = lim

ε→0E_G1_OJ,i(V_(i))(Φ^(l)(f^J(V_(i), b_i−ε))(g^J 1

∂vif^J)(V_(i), b_i−ε).

To shorten the notation we write simplyF(b_i−ε) =f^J(V_(i), b_i−ε).

Now one can prove that ifU ∈ S^l′(∪^l+1_k=1I_k) for 1≤l^′ ≤lthen∀i∈I_l+1,U(b_i−ε)∈ S^l′(∪^l_k=1I_k) and

|U(b_i−ε)|l^′ ≤ |U|l^′. We deduce then that∀i∈I_l+1F(b_i−ε)∈ S^l+1(∪^l_k=1I_k) and that (G_∂¹

viF)(b_i−ε)∈ S^l(∪^l_k=1I_k) and from induction hypothesis

|EG(Φ^(l)(F(b_i−ε))1_OJ,i(G_∂¹

viF)(b_i−ε)| ≤C_l||Φ||∞E_G|G(bi−ε)|l|_∂ ¹

viF(bi−ε)|l(1 +|p|0)^lΠ_l(F(b_i−ε)),

≤C_l||Φ||_∞E_G|G|_l|_∂¹

viF|_l(1 +|p|₀)^lΠ_l(F).

Putting this in (22) we obtain

|E_G X

i∈Il+1

1_Λl+1,i1_OJ,i(Φ^(l)(F)G 1

∂_viFp_i)(b⁻_i )| ≤ C_l||Φ||_∞E_G|G|_l(1 +|p|₀)^lΠ_l(F) X

i∈Il+1

1_Λl+1,i|p_i|_∞| 1

∂_viF|_l,

≤ C_l||Φ||_∞E_G|G|_l(1 +|p|₀)^l+1Π_l(F)|(D_l+1F)⁻¹|_l. (23) Finally plugging (21) and (23) in (20)

|EG(Φ^(l+1)(F)G)| ≤ C_l||Φ||∞ E_G|G|l+1(1 +|p|0)^lΠ_l+1(F) +E_G|G|l(1 +|p|0)^l+1Π_l(F)|(Dl+1F)⁻¹|l

,

≤ C_l||Φ||∞E_G|G|l+1(1 +|p|0)^l+1Π_l+1(F),

and inequality (17) is proved forl+ 1. This achieves the first part of the proof of Theorem 1.

It remains to prove (19). We assume thatω ∈ {γ >0}.

Let 1≤k≤l. We first notice that combining (13) and (14) we obtain

|δ_k(F)|_k−1 ≤ |F|_k(1 +

D_klnp_(k) ∞), sincep_(k) only depends on the variables Vi, i∈I_k. So we deduce the bound

δ_k((D_kF)⁻¹)

k−1 ≤

(D_kF)⁻¹

k(1 +|lnp|₁). (24) Now we have

|(DkF)⁻¹|k−1 =X

i∈I_k

1_Λk,i| 1

∂_viF|k−1

From (15) with φ(x) = 1/x

| 1

∂viF|k−1 ≤C_k(1 +|F|^k−1_k ) γ^k , and consequently

|(DkF)⁻¹|k−1≤C_k(1 +|F|^k−1_k )

γ^k . (25)

Moreover we have using successively (13) and (25)

|(D_kF)⁻¹|_k = |(D_kF)⁻¹|_k−1+|D_k(D_kF)⁻¹|_k−1,

≤ C_k

(1+|F|^k−1_k )

γ^k +^(1+|DkF|

k−1

k )

γ^k+1

,

≤ C_k^(1+|F|k−1k _γk+1^{+|DkF|k−1k )}. Putting this in (24)

δ_k((D_kF)⁻¹)

k−1≤C_k(1 +|F|^k−1_k +|DkF|^k−1_k )

γ^k+1 (1 +|lnp|₁). (26)

Finally from (25) and (26), we deduce

Π_l(F)≤C_l(1 +|lnp|₁)^l γ^l(l+2)

l

Y

k=1

(1 +|F|^k−1_k +|DkF|^k−1_k )²,

and Theorem 1 is proved. ⋄

3 Stochastic equations with jumps

3.1 Notations and hypotheses

We consider a Poisson point process p with measurable state space (E,B(E)). We refer to Ikeda and Watanabe [10] for the notation. We denote by N the counting measure associated to p so Nt(A) :=N((0, t)×A) = #{s < t;ps∈A}. The intensity measure is dt×dµ(a) whereµ is a sigma- finite measure on (E,B(E)) and we fix an non decreasing sequence (En) of subsets of E such that E =∪nEn,µ(En)<∞and µ(En+1)≤µ(En) +K for all nand for a constant K >0.

We consider the one dimensional stochastic equation X_t=x+

Z _t

0

Z

E

c(s, a, X_s⁻)dN(s, a) + Z _t

0

g(s, X_s)ds. (27)

Our aim is to give sufficient conditions on the coefficientscand gin order to prove that the law of X_t is absolutely continuous with respect to the Lebesgue measure and has a smooth density. We make the following assumptions on the coefficients c andg.

H1. We assume that the functionscandgare infinitely differentiable with respect to the variables (t, x) and that there exist a bounded functionc and a constantg, such that

∀(t, a, x) |c(t, a, x)| ≤c(a)(1 +|x|), sup

l+l^′≥1

|∂_t^l′∂_x^lc(t, a, x)| ≤c(a);

∀(t, x) |g(t, x)| ≤g(1 +|x|), sup

l+l^′≥1

|∂_t^l′∂_x^lg(t, x)| ≤g;

We assume moreover that R

Ec(a)dµ(a)<∞.

Under H1, equation (27) admits a unique solution.

H2. We assume that there exists a measurable function ˆc:E 7→R₊ such that R

Eˆc(a)dµ(a)<∞ and

∀(t, a, x) |∂_xc(t, a, x)(1 +∂_xc(t, a, x))⁻¹| ≤ˆc(a).

To simplify the notation we take ˆc =c. Under H2, the tangent flow associated to (27) is invertible.

At last we give a non-degeneracy condition wich will imply (16). We denote byα the function α(t, a, x) =g(t, x)−g(t, x+c(t, a, x)) + (g∂_xc+∂_tc)(t, a, x). (28) H3. We assume that there exists a measurable functionα:E 7→R₊ such that

∀(t, a, x) |α(t, a, x)| ≥α(a)>0,

∀n Z

En

1

α(a)dµ(a)<∞ and lim inf

n

1 µ(En)ln

Z

En

1

α(a)dµ(a)

=θ <∞.

We give in the following some examples whereE = (0,1] andα(a) =a.

3.2 Main results and examples

Following the methodology introduced in Bally and Cl´ement [2], our aim is to bound the Fourier transform ofX_t, ˆp_Xt(ξ), in terms of 1/|ξ|, recalling that ifR

R|ξ|^q|ˆp_Xt(ξ)|dξ <∞, forq >0, then the law of X_t is absolutely continuous and its density is C^[q]. This is done in the next proposition. The proof of this proposition relies on an approximation of X_t which will be given in the next section.

Proposition 3 Assuming H1, H2 and H3 we have for all n, L∈N^∗

|ˆp_Xt(ξ)| ≤C_t,L

e^{−µ(En)t/(2L)}+ 1

|ξ|^LA_n,L

, with A_n,L=µ(E_n)^L(R

En

1

α(a)dµ(a))^L(L+2).

From this proposition, we deduce our main result.

Theorem 2 We assume that H1, H2 and H3 hold. Let q ∈ N, then for t > 16θ(q+ 2)(q+ 1)², the law of Xt is absolutely continuous with respect to the Lebesgue measure and its density is of class C^q. In particular ifθ= 0, the law ofX_t is absolutely continuous with respect to the Lebesgue measure and its density is of class C^∞ for everyt >0.

Proof: From Proposition 3, we have

|ˆp_Xt(ξ)| ≤C_t,L

e^{−µ(En)t/2L}+ 1

|ξ|^LA_n,L

. Now∀k, k0 >0, ift/2L > kθ, we deduce from H3 that for n≥nL

t/2L > k µ(E_n)ln(

Z

En

1

α(a)dµ(a)) + klnµ(E_n) k₀µ(E_n) since the second term on the right hand side tends to zero. This implies

e^µ(En)t/2L >( Z

En

1

α(a)dµ(a))^kµ(E_n)^k/k0.

Choosingk=L(L+ 2) andk/k₀ =L, we obtain that forn≥n_Land t/2L > L(L+ 2)θ e^µ(En)t/2L> A_n,L.

and then

|ˆp_Xt(ξ)| ≤ C_t,L

e^{−µ(En)t/2L}+_|ξ|¹_Le^µ(En)t/2L ,

≤ C_t,L(_B¹

n(t) +^B_|ξ|^n(t)_L ),

with Bn(t) = e^µ(En)t/2L. Now recalling that µ(En) < µ(En+1) ≤ K +µ(En), we have Bn(t) <

B_n+1(t)≤K_tB_n(t). Moreover sinceB_n(t) goes to infinity with nwe have 1_{|ξ|L/2≥B_nL(t)} = X

n≥nL

1_{{Bn(t)≤|ξ|}L/2<Bn+1(t)}.

But ifB_n(t)≤ |ξ|^L/2< B_n+1(t), |ˆp_Xt(ξ)| ≤C_t,L/|ξ|^L/2 and so R |ξ|^q|ˆp_Xt(ξ)|dξ = R

|ξ|^L/2<B_nL(t)|ξ|^q|ˆp_Xt(ξ)|dξ+R

|ξ|^L/2≥B_nL(t)|ξ|^q|ˆp_Xt(ξ)|dξ,

≤ C_t,L,nL+R

|ξ|^L/2≥B_nL(t)|ξ|^q−L/2dξ.

For q ∈ N, choosing L such that L/2−q >1, we obtain R

|ξ|^q|ˆp_Xt(ξ)|dξ <∞ for t/2L > L(L+ 2)θ and consequently the law of X_t admits a density C^q for t > 2L²(L+ 2)θ and L > 2(q + 1), that is t >16θ(q+ 1)²(q+ 2) and Theorem 2 is proved.

⋄ We end this section with two examples

Example 1. We take E = (0,1], µ_λ = P

k≥1 1

k^λδ_1/k with 0 < λ < 1 and E_n = [1/n,1]. We have ∪nE_n = E, µ(E_n) = P_n

k=1 1

k^λ and µ_λ(E_n+1) ≤ µ_λ(E_n) + 1 . We consider the process (X_t)

solution of (27) withc(t, a, x) = aand g(t, x) =g(x) assuming that the derivatives of g are bounded and that |g^′(x)| ≥ g > 0. We have R

Eadµ_λ(a) = P

k≥1 1

k^λ+1 < ∞ so H1 and H2 hold. Moreover α(t, a, x) =g(x)−g(x+a) soα(a) =ga. Now R

En

1

adµ_λ(a) =Pn

k=1k^1−λ which is equivalent asngo to infinity ton^2−λ/(2−λ). Now we have

1 µ_λ(E_n)ln

Z

En

1

α(a)dµ_λ(a)

= ln(gP_n

k=1k^1−λ) P_n

k=1 1 k^λ

∼_n→∞Cln(n^2−λ) n^1−λ →0,

and then H3 is satisfied with θ = 0. We conclude from Theorem 2 that ∀t >0, X_t admits a density C^∞.

In the caseλ= 1, we haveµ₁(E_n) =P_n

k=1 1

k ∼_n→∞lnnthen 1

µ₁(E_n)ln Z

En

1

α(a)dµ₁(a)

= ln(gP_n

k=11) P_n

k=11 k

∼_n→∞ 1,

and this gives H3 withθ= 1. So the density ofX_t is regular as soon astis large enough. In fact it is proved in Kulik [12] that under some appropriate conditions the density of X_t is not continuous for small t.

Example 2. We take the intensity measure µ_λ as in the previous example and we consider the process (X_t) solution of (27) withg= 1 and c(t, a, x) =ax. This gives c(a) =aandα(a) =a. So the conclusions are similar to example 1 in both cases 0< λ <1 and λ= 1. But in this example we can compare our result to the one given by Ichikawa and Kunita [10]. They assume the condition

lim inf

u→0

1 u^h

Z

|a|≤u

a²dµ(a)>0, (⋆) for someh∈(0,2). Here we have

Z

|a|≤u

a²dµ(a) = X

k≥1/u

1

k^2+λ ∼_u→0 u^1+λ 1 +λ.

So if 0< λ <1, (⋆) holds and their results apply. In the caseλ= 1, (⋆) fails and they do not conclude.

However in our approach we conclude that the density of X_t is C^q for t >16(q+ 2)(q+ 1)². The next section is devoted to the proof of Proposition 3.

3.3 Approximation of Xt and integration by parts formula

In order to bound the Fourier transform of the processX_tsolution of (27), we will apply the differential calculus developed in section 2. The first step consists in an approximation ofX_tby a random variable

X_t^N which can be viewed as an element of our basic space S⁰. We assume that the process (X_t^N) is solution of the discrete version of equation (27)

X_t^N =x+ Z _t

0

Z

EN

c(s, a, X_s^N−)dN(s, a) + Z _t

0

g(s, X_s^N)ds. (29)

Since µ(E_N) < ∞, the number of jumps of the process X^N on the interval (0, t) is finite and con- sequently we may consider the random variable X_t^N as a function of these jump times and ap- ply the methodology proposed in section 2. We denote by (J_t^N) the Poisson process defined by J_t^N =N((0, t), E_N) = #{s < t;p_s ∈E_N}and we note (T_k^N)_k≥1 its jump times. We also introduce the notation ∆^N_k =p_TN

k . With these notations, the process solution of (29) can be written X_t^N =x+

J_t^N

X

k=1

c(T_k^N,∆^N_k, X_T^NN k −) +

Z _t

0

g(s, X_s^N)ds. (30)

We will not work with all the variables (T_k^N)_k but only with the jump times (T_kⁿ) of the Poisson process J_tⁿ, where n < N. In the following we will keep n fixed and we will make N go to infinity.

We note (T_k^N,n)_k the jump times of the Poisson processJ_t^N,n =N((0, t), E_N\En) and ∆^n,N_k =p_Tn,N

k .

Now we fixeL∈N^∗, the number of integration by parts and we notet_l =tl/L, 0≤l≤L. Assuming that J_tⁿ_l −J_tⁿ_l−1 =m_l for 1≤l≤L, we denote by (T_l,iⁿ)_1≤i≤ml the jump times of J_tⁿ belonging to the time interval (t_l−1, t_l). In the following we assume that m_l ≥1,∀l. For i= 0 we setT_l,0ⁿ =t_l−1 and fori=m_l+ 1, T_l,mⁿ

l+1 =t_l. With these definitions we choose our basic variables (V_i, i∈I_l) as (V_i, i∈I_l) = (T_l,2i+1ⁿ ,0≤i≤[(m_l−1)/2]). (31) The σ-algebra which contains the noise which is not involved in our differential calculus is

G =σ{(J_tⁿ_l)_1≤l≤L; (T_l,2iⁿ )_{1≤2i≤ml,1≤l≤L}; (T_k^N,n)_k; (∆^N_k )_k}. (32) Using some well known results on Poisson processes, we easily see that conditionally onGthe variables (V_i) are independent and for i∈I_l the law of V_i conditionally on G is uniform on (T_l,2iⁿ , T_l,2i+2ⁿ ) and we have

pi(v) = 1

T_l,2i+2ⁿ −T_l,2iⁿ 1_(Tⁿ

l,2i,T_l,2i+2ⁿ )(v), i∈I_l, (33)

Consequently taking a_i = T_l,2iⁿ and b_i = T_l,2i+2ⁿ we check that hypothesis H0 holds. It remains to define the localizing sets (Λ_l,i)_i∈Il.

We denote

hⁿ_l = t_l−t_l−1 2m_l = t

2Lm_l

and n_l= [(m_l−1)/2]. We will work on theG measurable set

Λⁿ_l =∪ⁿ_i=0^l {T_l,2i+2ⁿ −T_l,2iⁿ ≥hⁿ_l}, (34) and we consider the following partition of this set:

Λ_l,0 = {T_l,2ⁿ −T_l,0ⁿ ≥hⁿ_l},

Λ_l,i = ∩ⁱ_k=1{T_l,2kⁿ −T_l,2k−2ⁿ < hⁿ_l} ∩ {T_l,2i+2ⁿ −T_l,2iⁿ ≥hⁿ_l}, i= 1, ..., n_l.

After L−l iterations of the integration by parts we will work with the variables V_i, i ∈ I_l so the corresponding derivative is

D_lF =X

i∈Il

1_Λl,i∂_ViF =X

i∈Il

1_Λl,i∂_Tⁿ

l,2i+1F.

If we are on Λⁿ_l then we have at least one i such that t_l−1 ≤ T_l,2iⁿ < T_l,2i+1ⁿ < T_l,2i+2ⁿ ≤ t_l and T_l,2i+2ⁿ −T_l,2iⁿ ≥hⁿ_l.Notice that in this case 1_Λl,i|p_i|_∞≤(hⁿ_l)⁻¹ and roughly speaking this means that the variable V_i =T_l,2i+1ⁿ gives a sufficiently large quantity of noise. Moreover, in order to perform L integrations by parts we will work on

Γⁿ_L=∩^L_l=1Λⁿ_l (35)

and we will leave out the complementary of Γⁿ_L.The following lemma says that on the set Γⁿ_Lwe have enough noise and that the complementary of this set may be ignored.

Lemma 1 Using the notation given in Theorem 1 one has i)|p|₀ := max_1≤l≤LP

i∈Il1_Λl,i|pi|_∞≤ ^2L_t J_tⁿ, ii)P((Γⁿ_L)^c)≤Lexp(−µ(En)t/2L).

Proof: As mentioned before 1_Λl,i|pi|_∞ ≤ (hⁿ_l)⁻¹ = 2Lm_l/t ≤ ^2L_t J_tⁿ and so we have i). In order to prove ii) we have to estimate P((Λⁿ_l)^c) for 1 ≤ l ≤ L. We denote s_l = ¹₂(t_l +t_l−1) and we will prove that {J_tⁿ_l−J_sⁿ_l ≥1} ⊂Λⁿ_l. Suppose first that m_l =J_tⁿ_l −J_tⁿ_l−1 is even. Then 2n_l+ 2 = m_l. If T_l,2i+2ⁿ −T_l,2iⁿ < hⁿ_l for every i= 0, ..., n_l then

T_l,mⁿ _l−t_l−1 =

n_l

X

i=0

(T_l,2i+2ⁿ −T_l,2iⁿ )≤(n_l+ 1)× t

2Lm_l ≤ t

4L ≤s_l−t_l−1