Existence of densities for jumping S.D.E.s

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Existence of densities for jumping S.D.E.s

Nicolas Fournier, Jean-Sébastien Giet

To cite this version:

Nicolas Fournier, Jean-Sébastien Giet. Existence of densities for jumping S.D.E.s. Stochastic Processes

and their Applications, Elsevier, 2005, 116 (4), pp.643-661. �hal-00140440�

Existence of densities for jumping S.D.E.s

Nicolas Fournier and Jean-S´ ebastien Giet

Institut Elie Cartan, Campus Scientifique, BP 239 54506 Vandoeuvre-l`es-Nancy Cedex, France fournier@iecn.u-nancy.fr, giet@iecn.u-nancy.fr

September 2, 2005

Abstract

We consider a jumping Markov process{X_t^x}t≥0. We study the absolute continuity of the law of X_t^x fort >0. We first consider, as Bichteler-Jacod [2] and Bichteler-Gravereaux-Jacod [1], the case where the rate of jump is constant. We state some results in the spirit of those of [2, 1], with rather weaker assumptions and simpler proofs, not relying on the use of stochastic calculus of variations.

We finally obtain the absolute continuity of the law ofX_t^xin the case where the rate of jump depends on the spatial variable, and this last result seems to be new.

Key words: Stochastic differential equations, Jump processes, Absolute continuity.

MSC 2000: 60H10, 60J75.

1 Introduction

Consider ad-dimensional Markov process with jumps {X_t^x}^t≥0, starting fromx∈R^d, with generatorL, defined forφ:R^d7→Rsufficiently smooth andy∈R^d, by

Lφ(y) =b(y).∇φ(y) + Z

Rⁿ

γ(y)ϕ(z) [φ(y+h(y, z))−φ(y)]dz, (1.1)

with possibly an additional diffusion term, and the integral part written in a (more general)compensated form. Heren∈Nis fixed, and the functions γ:R^d7→Rand ϕ:Rⁿ7→Rare nonnegative.

We aim to investigate the absolute continuity of the law ofX_t^xwith respect to the Lebesgue measure on R^d, fort >0. We will sometimes allow the presence of a Brownian part, but we will actually not use the regularizing effect of the Brownian motion.

Assume for a moment thatd=n= 1. Roughly speaking, the law ofX_t^xis expected to have a density as soon ast >0, if for ally∈R,γ(y)R

Rϕ(z)dz=∞and ifh(y, z) is not too much constant inz(for example h(y, .) of classC¹with a nonzero derivative almost everywhere). Indeed, in such a case,X^xhas infinitely many jumps immediately aftert= 0. Furthermore, the jumps are of the shape X_t^x=X_t^x₋+h(X_t^x₋, Z), withZ a random variable independent ofX_t^x₋, with law ϕ(z)dz. This produces absolute continuity for X_t^x, ifhis sufficiently non-constant inz.

This simple idea is not so easy to handle rigorously, sinceX^xhas infinitely many jumps, and sinceϕ(z)dz is not a probability measure (because R

Rⁿϕ(z)dz = ∞). To our knowledge, all the known results are based on the use ofstochastic calculus of variations, i.e. on a sort ofdifferential calculuswith respect to the stochastic variableω. The first results in this direction were obtained by Bismut [3]. Important results are due to Bichteler-Jacod [2], and then Bichteler-Gravereaux-Jacod [1]. We refer to Graham-M´el´eard [7] and Fournier-Giet [6] for applications to physical integro-differential equations such as the Boltzmann and the coagulation-fragmentation equations. See also Picard [11] and Denis [4] for alternative methods in the much more complicated case where the intensity measure ofN is singular.

All the previously cited works concern the case where therate of jumpγ(x) is constant. The case where γis non constant is much more delicate. The main reason for this is that in such a case, the mapx7→X_t^x cannot be regular (and even continuous). Indeed, ifγ(x)< γ(y), and ifR

Rⁿϕ(z)dz=∞, then it is clear that for all smallt >0,X^y jumps infinitely more often thanX^xbeforet. The only available result with γnot constant seems to be that of [5], of which the assumptions are very restrictive: monotonicity (inx) is assumed forhandγ.

First, we would like to give some results in the spirit of Bichteler-Jacod [2] and Bichteler-Gravereaux- Jacod [1], with simpler proofs. We will in particular not use the stochastic calculus of variations. We thus consider in Section 2 the case where γ is constant. In Subsection 2.1, we state and prove a first result under a strong nondegeneracy assumption onh, which is satisfying only in the case whered= 1.

It relies on assumptions which ensure that one jump is sufficient to produce absolute continuity for the law of X_t^x. The proof is elementary, and our result follows the line of Theorem 2.5 in Bichteler-Jacod [2], but our assumptions are rather weaker. In Subsection 2.2, we study a much more complicated case, where a finite number of jumps are required to ensure the absolute continuity of the law of X_t^x. Our result is inspired by that of Bichteler-Gravereaux-Jacod [1] Theorem 2.14.

Although the results of Subsection 2.1 are contained in those of Subsection 2.2, we begin with Subsection 2.1 for the sake of clarity: the result and its proof are much simpler.

We will finally obtain a result in the case whereγ is not constant in Section 3. This last result seems to be new, and improves consequently those of [5].

Our methods allow to improve slightly the results of [1, 2] concerning the existence of a density whenγ is constant, and to obtain a result whenγ depends on the variable position. Let us however recall that the smoothness of the density was studied in [1], which does not seem to be possible with our method.

In the whole paper, we denote the collection of Borelian subsets ofR^d with vanishing Lebesgue measure by

A=

A∈ B(R^d);

Z

A

dx= 0

. (1.2)

2 The case of a constant rate of jump

Consider the followingd-dimensional S.D.E., for somed∈N, starting fromx∈R^d: X_t^x=x+

Z t 0

b(X_s^x)ds+ Z t

0

Z

Rⁿ

h(X_s^x₋, z) ˜N(ds, dz) + Z t

0

σ(X_s^x)dBs, (2.1)

where

Assumption (I): N(ds, dz) is a Poisson measure on [0,∞)×Rⁿ, for some n∈N, with intensity mea- sure ν(ds, dz) = dsϕ(z)dz. The function ϕ : Rⁿ 7→ R₊ is supposed to be measurable. We denote by N˜ =N−ν the associatedcompensatedPoisson measure. The R^m-valued (for some m∈N) Brownian motion{Bt}^t≥0is supposed to be independent ofN.

In this case, the generator of the Markov processX^xis given, for anyφ∈C_b²(R^d) andy∈R^d, by Lφ(y) =b(y).∇φ(y) +1

2

d

X

i,j=1

(σσ^∗)i,j(x)∂i∂jφ(x) + Z

Rⁿ

[φ(y+h(y, z))−φ(y)−h(y, z).∇φ(y)]ϕ(z)dz.

(2.2) We assume the following hypothesis,M^d×m standing for the set ofd×mmatrices with real entries.

Assumption(H1): The functionsb:R^d 7→R^d andσ:R^d 7→ M^d×mare of classC² and have at most linear growth. The function h:R^d×Rⁿ 7→R^d is measurable. For each z∈Rⁿ, x7→h(x, z) is of class C² on R^d. There exists η ∈ L²(Rⁿ, ϕ(z)dz) and a continuous function ζ : R^d 7→ R such that for all x∈R^d, z∈Rⁿ,|h(x, z)| ≤(1 +|x|)η(z), while|h⁰_x(x, z)|+|h⁰⁰_xx(x, z)| ≤ζ(x)η(z).

Then it is well-known that the following result holds.

Proposition 2.1 Assume (I) and (H1). Consider the natural filtration {F^t}^t≥0 associated with the Poisson measure N and the Brownian motion B. Then, for any x∈ R^d, there exists a unique c`adl`ag {F^t}^t≥0-adapted process {X_t^x}^t≥0 solution to (2.1) such that for allx∈R^d, allT ∈[0,∞),

E

"

sup

s∈[0,T]|X_s^x|²

#

<∞. (2.3)

The process{X_t^x}t≥0,x∈R^d furthermore satisfies the strong Markov property.

See Ikeda-Watanabe [8] for the case of globally Lipschitz coefficients. A standard localization procedure allows to obtain Theorem 2.1.

2.1 Absolute continuity using one jump

We first give some assumptions, statements, and examples. The proof is handled in a second part.

2.1.1 Statements

We first introduce some assumptions. HereId stands for the unitd×dmatrix, whilex0 is a given point ofR^d.

Assumption(H2): For allx∈R^d, allz∈Rⁿ, det(Id+h⁰_x(x, z))6= 0.

Assumption(H3)(x0): There exists >0 such that for allx∈B(x0, ), there exists a subsetO(x)⊂Rⁿ such that, (recall (1.2)),

Z

O(x)

ϕ(z)dz=∞, and for allA∈ A, Z

O(x)

1_{h(x,z)∈A}ϕ(z)dz= 0, (2.4)

and such that the map (x, z)7→1_{z∈O(x)} is measurable onB(x0, )×Rⁿ.

The main results of this section are the following.

Theorem 2.2 Letx0∈R^d be fixed. Assume(I),(H1),(H2)and(H3)(x0). Consider the unique solution {X_t^x0}^t≥0 to (2.1). Then the law of X_t^x0 has a density with respect to the Lebesgue measure on R^d as soon ast >0.

In the case where (H3)(x) holds for allx∈R^d, we can omit assumption (H2).

Corollary 2.3 Letx0∈R^d be fixed. Assume(I),(H1)and that (H3)(x)holds for allx∈R^d. Consider the unique solution {X_t^x0}^t≥0 to (2.1). Then the law of X_t^x0 has a density with respect to the Lebesgue measure onR^d as soon as t >0.

We do not state a result concerning the regularizing effect of the Brownian part of (2.1), since it seems reasonable that standard techniques of Malliavin calculus (see e.g. Nualart, [10]) may allow to prove that under (H1),(H2) and ifσσ^∗(x0) is invertible, then the law ofX_t^x0 has a density with respect to the Lebesgue measure onR^d as soon ast >0.

These results relax subsequently the assumptions of [2], especially concerning the regularity (inz) and boundness conditions on h. Let us comment on our hypotheses. The second condition in (2.4) means that the image measure of1_{z∈O(x)}ϕ(z)dz (where dz stands for the Lebesgue measure on Rⁿ) by the mapz7→h(x, z) has a density with respect to the Lebesgue measure onR^d, for eachx∈B(x0, ).

Proposition 2.4 Assume that n=d and that there exists an open subset O of Rⁿ such that (x, z)7→

h(x, z) is of class C¹ onR×O. If for all x∈R^d, R

O1_{deth⁰_z(x,z)6=0}ϕ(z)dz=∞, then (H3)(x0)holds for anyx0.

Indeed, it suffices to note that, since n = d, h⁰_z(x, z) is ad×d matrix for each x ∈ R^d, eachz ∈ O.

Choosing= 1 andO(x) ={z∈O, deth⁰_z(x, z)6= 0}allows us to conclude, noting that, due to the local inverse Theorem, the mapz7→h(x, z) is a localC¹-diffeomorphism onO.

Assumptions (H1) and (H3)(x0) are quite natural. Note that (H2) is not only a technical condition, as this example shows.

Example 2.5 Assume that n=d= 1, that ϕ≡1, that b, σsatisfy (H1) withb(0) =σ(0) = 0, and that h(x, z) =−x1_{|z|≤1}+(x/|z|)1_{|z|>1}. Then(I)and(H1)are satisfied, while(H3)(x)holds for allx6= 0, but (H2) fails. One can prove that in such a case, P[X_t^x0 = 0]> 0 for all t >0, and thus the law of X_t^x0 is not absolutely continuous. Indeed, it is clear that, ifT1= inf{t≥0; Rt

0

R

R1_{|z|≤1}N(ds, dz)≥1}, then T1 <∞ a.s. (its law is exponential with parameter 2) and XT1 = XT1−+ (−XT1−) = 0. Since furthermore b(0) = σ(0) = 0 and h(0, .) = 0, an uniqueness argument and the strong Markov property show thatXT1+t= 0a.s. for all t≥0. HenceP[X_t^x0= 0]>0 for allt >0.

2.1.2 Proof

We now turn to the proof of Theorem 2.2. We first proceed to a localization procedure.

Lemma 2.6 To prove Theorem 2.2 and Corollary 2.3, we may assume the additional condition (H4) below.

Assumption(H4): The functionsb, b⁰, b⁰⁰, σ, σ⁰, σ⁰⁰ are bounded. There exists ˜η∈L²(Rⁿ, ϕ(z)dz) such that for allx∈R^d,z∈Rⁿ,|h(x, z)|+|h⁰_x(x, z)|+|h⁰⁰_xx(x, z)| ≤η(z).˜

ProofWe study the case of Theorem 2.2. Letx0∈R^d be fixed. Assume that Theorem 2.2 holds under (I), (H1), (H2), (H3)(x0), (H4), and consider some functions b, σ, h satisfying only (I), (H1), (H2), (H3)(x0). For each l ≥ 1, consider some functions bl, σl, hl satisfying (I), (H1), (H2), (H3)(x0), (H4) and such that for all |x| ≤ l, all z∈Rⁿ, bl(x) =b(x), σl(x) =σ(x), andhl(x, z) =h(x, z). Denote by {X_t^x0,l}^t≥0 the solution to (2.1) withh, σ, b replaced byhl, σl, bl. Then, by assumption, the law of X_t^x0,l has a density ift >0. Next, denote byτl= inf{t≥0, |X_t^x0| ≥l}. It is clear from (2.3) thatτl increases a.s. to infinity as l tends to infinity. Furthermore a uniqueness argument yields that a.s. X_t^x0 =X_t^x0,l for allt≤τl. Hence, for anyA∈ A, anyt >0, by the Lebesgue Theorem,

P[X_t^x0∈A] = lim

l→∞P[X_t^x0 ∈A, t < τl] = lim

l→∞P[X_t^x0,l ∈A, t < τl]≤ lim

l→∞P[X_t^x0,l ∈A] = 0, (2.5) since the law ofX_t^x0,l has a density for eachl≥1. This implies that the law of X_t^x0 has a density.

We now gather some known results about theflow{X_t^x}t≥0,x∈R^d.

Lemma 2.7 Assume(I),(H1),(H4). Consider the flow of solutions{X_t^x}t≥0,x∈R^d to (2.1). Then a.s., the mapx7→X_t^xis of classC¹ onR^d for eacht≥0. If furthermore(H2)holds, then a.s., for all t≥0, allx∈R^d,det_∂x^∂ X_t^x6= 0.

ProofIt is well-known (see Protter [13] Theorems 39 and 40 Section 7 for very similar results) that under (I),(H1),(H4), the mapx7→X_t^xis a.s. of classC¹ onR^d for eacht≥0, and that one may differentiate (2.1) with respect tox:

∂

∂xX_t^x=Id+ Z t

0

b⁰(X_s^x) ∂

∂xX_s^xds+ Z t

0

Z

Rⁿ

h⁰_x(X_s^x₋, z) ∂

∂xX_s^x₋N(ds, dz) +˜ Z t

0

σ⁰(X_s^x) ∂

∂xX_s^xdBs. (2.6) Then, following the ideas of Jacod ([9], Theorem 1 and Corollary page 443), we deduce an explicit expression forV_t^x= det_∂x^∂ X_t^xin terms of Dol´eans-Dade exponentials (a continuity argument shows that this explicit expression holds a.s. simultaneously for allx∈R^d). We thus obtain, still using the results of [9] simultaneously for allx∈R^d, that a.s., det_∂x^∂ X_t^x6= 0 for allx∈R^d and allt < τ = infx∈R^dT^x, where

T^x= inf{t≥0;

Z t 0

Z

Rⁿ

1_{det(Id+h⁰

x(X_s−^x ,z))=0}N(ds, dz)≥1}. (2.7)

But (H2) ensures thatτ=∞a.s.

We may now prove Theorem 2.2.

Proof of Theorem 2.2 Due to Lemma 2.6, we suppose the additional condition (H4). We consider x0∈R^d andt >0 fixed.

Step 1: Due to (H3)(x0), we may build, for eachx∈ B(x0, ), an increasing sequence {Op(x)}^p≥1 of subsets ofRⁿ such that

∪^p≥1Op(x) =O(x) and∀p≥1, Z

Op(x)

ϕ(z)dz=p, (2.8)

in such a way that for eachp≥1, the map (x, z)7→1_{z∈Op(x)}is measurable onB(x0, )×Rⁿ. We also consider the stopping time

τ = inf{s≥0; |X_s^x0−x0| ≥}>0 a.s. (2.9)

The positivity ofτ comes from the fact thatX^x0 is a.s. right-continuous and starts fromx0. We finally consider the stopping time, forp≥1,

Sp= inf

s≥0;

Z s 0

Z

Rⁿ

1ⁿ

z∈Op

“

X^x_(u∧τ)−⁰ ”oN(du, dz)≥1

, (2.10)

and the associatedmarkZp∈Rⁿ, uniquely defined byN({Sp} × {Zp}) = 1.

Due to (2.8), and to the fact thatX_(u^x0_∧τ)− always belongs toB(x0, ), one may prove that (i)p7→Sp is a.s. nonincreasing,

(ii) limp→∞Sp= 0 a.s.,

(iii) conditionally toF^Sp−, the law ofZp is given by ¹_pϕ(z)1ⁿ

z∈Op

“X^x₍⁰

Sp∧τ)−

”odz, where

F^Sp− =σ{B∩ {Sp> s}; s≥0, B∈ F^s}. (2.11)

Indeed, (i) is obvious by construction, sincep7→Op(x) is increasing for eachx∈R^d. Next, an easy compu- tation shows that the compensator of the (random) point measureN^p(ds, dz) =1_{z∈O

p(X_(s∧τ)−^x0 )}N(ds, dz) is given bypds×p⁻¹1_{z∈Op(X^x0

(s∧τ)−)}ϕ(z)dz. Since for each x∈ R^d, R

Rⁿp⁻¹1_{z∈Op(x)}ϕ(z)dz = 1, we deduce that the rate of jump of N^p is constant and equal to p, so thatSp, which is its first instant of jump, has an exponential distribution with parameterp. This and (i) ensure (ii). We also obtain (iii) as a consequence of the shape of the compensator ofN^p.

Step 2: We now prove that conditionally to σ(Sp), the law of X_S^x0_p has a density with respect to the Lebesgue measure on R^d, on the set Ω⁰_p = {τ ≥ Sp}. Since Sp is F^Sp−-measurable, it suffices to prove that for any A ∈ A, a.s., P[Ω⁰_p, X_S^x_p⁰ ∈ A | F^Sp−] = 0. But, using the notations of Step 1, a.s., X_S^x0_p=X_S^x0_p−+h[X_S^x0_p−, Zp] on Ω⁰_p. Furthermore, we know that on Ω⁰_p,X_S^x_p⁰₋ ∈B(x0, ) a.s. Thus, using

Step 1 (see (iii)), since{τ ≥Sp}andX_S^x_p⁰₋ areF^Sp−-measurable, P[Ω⁰_p, X_S^x0_p∈A| F^Sp−] = 1_Ω^p

0Ph

X_S^x_p⁰₋+h[X_S^x_p⁰₋, Zp]∈A| F^Sp−

i (2.12)

= 1ⁿ

τ≥S_p,X^x0

Sp−∈B(x0,)o

Z

Rⁿ

1ⁿ

hh X^x0

Sp−,zi

∈A−X^x0

Sp−

o

1

pϕ(z)1ⁿ_z

∈O_p“ X^x0

Sp−

”odz= 0 due to (H3)(x0) (use (2.4) withx=X_S^x_p⁰₋), since for anyy∈R^d,A−y={x−y, x∈A} belongs toA.

Step 3: We may now deduce that for anyp≥1, the law ofX_t^x0has a density on the set Ω¹_p={Sp≤τ∧t}. We deduce from Step 2 that on Ω¹_p ⊂Ω⁰_p the law of (Sp, X_S^x0_p) is of the shape νp(ds)fp(s, x)dx. Hence, for anyA∈ A, using the strong Markov property, we obtain, conditionning with respect toF^Sp,

P[Ω¹_p, X_t^x0∈A] = E

1_Ω¹_pE Z t

0

νp(ds) Z

R^d

fp(s, x)dx1_{X_t−s^x ∈A}

. (2.13)

It thus suffices to show that a.s., for anys < t fixed, Z

R^d

fp(s, x)dx1_{X_t−s^x ∈A}= 0. (2.14)

Of course, it suffices to check that a.s., fors < t fixed, Z

R^d

dx1_{X_t−s^x ∈A}= 0. (2.15)

But this is immediate from Lemma 2.7, using that the Jacobian of the mapx7→X_t^x_−sdoes (a.s.) never vanish and that Ais Lebesgue-nul: one may find, due to the local inverse Theorem, a countable family of open subsetsRi ofR^d, on whichx7→X_t^x_−sis aC¹diffeomorphism, and such thatR^d=∪^∞i=1Ri. The conclusion follows, performing the substitutionx7→ y =X_t^x_−s on each Ri. This allows us to conclude thatP[Ω^p₁, X_t^x0∈A] = 0.

Step 4: The conclusion readily follows: due to Step 1 (see (2.9) and (ii)),1_Ω¹_p goes a.s. to 1 asptends to infinity. We thus infer from the Lebesgue Theorem that for anyA∈ A,

P[X_t^x0∈A] = lim

p→∞P[Ω¹_p, X_t^x0∈A] = 0, (2.16) thanks to Step 3. Thus the law ofX_t^x0 has a density with respect to the Lebesgue measure onR^d.

We finally show how to relax assumption (H2) when (H3)(x) holds everywhere.

Proof of Corollary 2.3 Due to Lemma 2.6, we may suppose the additional assumption (H4). We considerδ0>0 such that for anyM ∈ M^d×d satisfying|M| ≤δ0, det(Id+M)≥1/2. We then splitRⁿ intoOF ∪OI, with

OF ={z∈Rⁿ, η(z)˜ ≥δ0}, OI ={z∈Rⁿ, η(z)˜ < δ0}. (2.17)

Since ˜η ∈ L²(Rⁿ, ϕ(z)dz), we deduce that λF := R

OF ϕ(z)dz ≤ δ⁻₀²R

Rⁿη˜²(z)ϕ(z)dz < ∞. We next consider the solution{Y_t^x}^t≥0to the S.D.E.

Y_t^x=x+ Z t

0

b(Y_s^x)ds+ Z t

0

Z

Rⁿ

h(Y_s^x₋, z)1_{z∈OI}N˜(ds, dz)− Z t

0

Z

OF

h(Y_s^x₋, z)ϕ(z)dz+ Z t

0

σ(Y_s^x)dBs. (2.18) Clearly, this S.D.E. satisfies (I), (H1), (H2), and (H3)(x) for allx, so that due to Theorem 2.2, the law ofY_t^x has a density for eacht >0, eachx∈R^d. The solutionX_t^x0 to (2.1) may now be realized in the following way (see Ikeda-Watanabe [8] for details): consider a standard Poisson process with intensity λF and with instants of jump 0 =T0 < T1 < T2 < ..., a family of i.i.d. Rⁿ-valued random variables (Zi)i≥1 with law λ⁻_F¹ϕ(z)1_{z∈OF}dz, and a family of i.i.d. solutions ({Y_t^i,x}x∈R^d,t≥0)i≥1 to (2.18), all these random objects being independent. Set

X₀^x0 =x0, ∀i≥0, X_T^x_i+1⁰ =Y^i,X

x0 Ti

Ti+1−Ti+h(Y^i,X

x0 Ti

Ti+1−Ti, Zi) and∀t≥0, X_t^x0=X

i≥0

Y^i,X

x0 Ti

t−Ti 1_{{t∈[Ti,Ti+1)}}. (2.19) Then{X_t^x0}^t≥0 is solution (in law) to (2.1). To conclude, notice that for anyt >0, one has t /∈ ∪ⁱ{Ti} a.s., so that for anyA∈ A,

P[X_t^x0∈A] = X

i≥0

P

Y^i,X

x0 Ti

t−Ti ∈A, t∈(Ti, Ti+1)

≤X

i≥0

Ph

Y_t^i,X_−T^Ti_i ∈A, t > Ti

i= 0. (2.20)

The last equality comes from the facts that for each i, {Y_s^i,x}s≥0,x∈R^d is independent of (Ti, XTi), and that the law ofY_t^i,xhas a density for eacht >0, eachx∈R^d.

2.2 Absolute continuity using a finite number of jumps

We now would like to investigate the case where one jump is not sufficient to produce a density for the law ofX_t^x. For example, assume thatd= 2, and thatX_t^x= (X_t^x,1, X_t^x,2) has sufficiently many jumps which produce a density forX_t^x,1, sufficiently many jumps which produce a density forX_t^x,2, and that these two kinds of jump are independent enough. Then the result of Theorem 2.2 might still hold, even if (H3) fails.

In other words, we would like to prove a result in the spirit of Bichteler-Gravereaux-Jacod [1] Theorem 2.14. We keep in mind assumptions (I), (H1), (H2) and (H4) defined in the previous subsection.

We first give our result, which relies on a general (but quite untracktable) non-degeneracy assumption.

Then we turn to the proof, and we conclude the section with examples of applications.

2.2.1 Statements

We invite the reader to have a look at Assumption (H6) (stated in Subsection 2.2.3), which is a tractable version of (H5) below ((H6) is however less general). We first of all introduce some notation.

Notation 2.8 Assume(I)and(H1).

(i)Forx= (xi)∈R^d, we set |x|= (P

ix²_i)^1/2, and for M = (Mij)i,j∈ M^d×d we set |M|=P

i,j|Mij|. (ii)Forx∈R^d and >0, we consider the set

Ox, ={z∈Rⁿ, |h(x, z)| ≤}. (2.21)

(iii)Forx∈R^d,η >0and >0, we consider the following set of regular functions:

D^x,η,= (

ψ∈C¹(B(x, η)7→R^d), sup

y∈B(x,η)

[|ψ(y)−y|+|ψ⁰(y)−Id|]≤ )

. (2.22)

Note that for anyψ inD^y,η,,ψ⁰(y)is ad×dmatrix for eachy∈B(x, ), and that one hasψ(B(x, η))⊂ B(x, η+).

(iv)We consider the functiong:R^d×Rⁿ 7→R^d defined by g(x, z) =x+h(x, z).

(v) Consider x0 ∈R^d, >0 and α∈N to be fixed. For ψ1, ..., ψα∈ D^x0,2α,, we define the functions

g_i^{x0,ψ1,...,ψi} for i∈ {1, ..., α} recursively, by

g^x₁^0,ψ1 : Oψ1(x0),⊂Rⁿ7→B(x0,2)⊂R^d,

g^x₁^0,ψ1(z1) = g[ψ1(x0), z1], (2.23)

and, fori∈ {1, ..., α−1},

g_i+1^{x0,ψ1,...,ψi+1}(z1, ..., zi, .) : O_ψ

i+1(g^x_i^{0,ψ1,...,ψi}(z1,...,zi)),⊂Rⁿ7→B(x0,2(i+ 1))⊂R^d, g_i+1^{x0,ψ1,...,ψi+1}(z1, ..., zi, zi+1) = g[ψi+1(g_i^{x0,ψ1,...,ψi}(z1, ..., zi)), zi+1]. (2.24)

Note that (2.23) and (2.24) always make sense, sinceψi belongs to D^x0,2α, and due to point (ii). For x0∈R^d to be fixed, our non-degeneracy assumption is the following.

Assumption (H5)(x0): There exists > 0 and α ∈ N such that for any ψ1, ..., ψα in D^x0,2α,, the following conditions hold: there existsO1(ψ1)⊂Oψ1(x0),such that

Z

O1(ψ1)

ϕ(z)dz=∞, (2.25)

and such that for allz1∈O1(ψ1), there existsO2(ψ1, ψ2;z1)⊂O_ψ

2[g^x₁^0,ψ1(z1)],such that Z

O2(ψ1;ψ2,z1)

ϕ(z)dz=∞, ..., (2.26)

and such that for allzα−1∈Oα−1(ψ1, ..., ψα−1;z1, ..., zα−2), there exists Oα(ψ1, ..., ψα;z1, ..., zα−1)⊂O_ψ

α[g_α−1^{x0,ψ1,...,ψα−1}(z1,...,zα−1)],such that Z

Oα(ψ1,...,ψα;z1,...,zα−1)

ϕ(z)dz=∞, (2.27)

and such that for all negligible Lebesgue subsetA∈ A(recall (1.2)), Z

O1(ψ1)

dz1

Z

O2(ψ1,ψ2;z1)

dz2...

Z

Oα(ψ1,...,ψα;z1,...,zα−1)

dzα1ⁿ

g_α^{x0,ψ1,...,ψα}(z1,...,zα)∈Ao= 0. (2.28) We finally require that for alli∈ {1, ..., α}, the map

ψ1, ..., ψi, z1, ..., zi7→1_{{zi∈Oi(ψ1,...,ψi;z1,...,zi−1)}} (2.29)

is measurable with respect to all its variables.

We then have the following results, which generalize Theorem 2.2 and Corollary 2.3.

Theorem 2.9 Letx0∈R^d be fixed. Assume(I),(H1),(H2)and(H5)(x0). Consider the unique solution {X_t^x0}^t≥0 to (2.1). Then the law of X_t^x0 has a density with respect to the Lebesgue measure on R^d as soon ast >0.

Corollary 2.10 Letx0∈R^dbe fixed. Assume(I),(H1), and that(H5)(x)holds for allx∈R^d. Consider the unique solution {X_t^x0}^t≥0 to (2.1). Then the law of X_t^x0 has a density with respect to the Lebesgue measure onR^d as soon as t >0.

The idea of Theorem 2.9 is relatively simple, since assumption (H5)(x0) almost contains its proof (see Subsection 2.2.2 below). Note thatαstands for themaximum number of jumpsnecessary to produce a density for the law ofX_t^x.

2.2.2 Proof

We first of all remark that we may assume the additionnal assumption (H4) as before: copy line by line the proof of Lemma 2.6. By the same way, the proof of Corollary 2.10 is the same as that of Corollary 2.3, using of course Theorem 2.9 instead of Theorem 2.2. We will need the following result concerning the flow{X_t^x}t≥0,x∈R^d.

Lemma 2.11 Assume(I),(H1) and(H4). Consider the flow of solutions{X_t^x}t≥0,x∈R^d to (2.1). Then for anyx0∈R^d,

limt→0E

"

sup

s∈[0,t]

sup

x∈B(x0,1)

(

|X_s^x−x|²+

∂X_s^x

∂x −Id

2)#

= 0. (2.30)

ProofFirst, a standard computation using (2.1), (2.6) and (H4) shows that there exists a constantC >0 such that for allx∈R^d, allt∈[0,1],

E

"

sup

s∈[0,t]

(

|X_s^x−x|²+

∂X_s^x

∂x −Id

2)#

≤Ct. (2.31)

Another standard computation using (2.1), (2.6) and (H4) shows that there exists a constantC >0 such that for allx, y∈R^d,

E

"

sup

s∈[0,1]

(

|X_s^x−X_s^y|²+

∂X_s^y

∂x −∂X_s^x

∂x

2)#

≤C|x−y|². (2.32)

We deduce from an easy adaptation of the Kolmogorov criterion of continuity, see e.g. Revuz-Yor [12] p 25, that there existsC >0 such that

ifZ= sup

s∈[0,1]

sup

x,y∈B(x0,1),x6=y







|X_s^x−X_s^y|²

|x−y|^1/2 +

∂X_s^y

∂x −^∂X∂x^sx

2

|x−y|^1/2







, E[Z]<∞. (2.33)

To conclude, consider, for each n ∈ N, some points {xi}ⁱ∈{1,...,n^d} of B(x0,1) such that for any x ∈ B(x0,1), mini|x−xi| ≤1/n. We then obtain, using (2.31) and (2.33), for someC not depending onn,

E

"

sup

s∈[0,t]

sup

x∈B(x0,1)

(

|X_s^x−x|²+

∂Xs^x

∂x −Id

2)#

≤E





n^d

X

i=1

sup

s∈[0,t]

(

|X_s^xi−xi|²+

∂X_s^xi

∂x −Id

2)

+n^−1/2Z



≤C[n^dt+n^−1/2] (2.34) One easily concludes, choosingnto be equal to the integer part oft^−1/2d. Proof of Theorem 2.9As said before, we may assume the additional condition (H4). Lett >0 and x0∈R^dbe fixed. We assume that (H5)(x0) is satisfied, for some >0. We supposesmaller than 1/2α, whereα∈Nwas defined in (H5)(x0). This ensures thatB(x0,2α)⊂B(x0,1). We denote, forx∈R^d ands≥0, by{X_r^x,s}^r≥sthe process defined by

X_r^x,s=x+ Z r

s

b(X_u^x,s)du+ Z r

s

Z

Rⁿ

h(X_u^x,s₋, z) ˜N(du, dz) + Z r

s

σ(X_u^x,s)dBu, (2.35)

We consider, for 0≤s≤r, the (stochastic) mapsψs,randψs,r− from R^d into itself, defined by

ψs,r(x) =X_r^x,s, ψs,r−(x) =X_r^x,s₋. (2.36)

Step 1: We first build recursively some well-chosen jump times and marks. Due to (H5)(x0), we may build, for anyψ1∈ D^x0,2α,, an increasing sequence{O^p₁(ψ1)}^p≥1 of subsets ofRⁿ such that

∪^p≥1O₁^p(ψ1) =O1(ψ1), and∀p≥1, Z

O^p₁(ψ1)

ϕ(z)dz=p. (2.37)

We also consider the stopping times

τ1 = inf{s≥0; ψ0,s∈ D/ ^x0,2α,}, S^p₁ = inf

s≥0;

Z s 0

Z

Rⁿ

1_{z∈O^p

1(ψ0,(u∧τ1 )−)}N(du, dz)≥1

, (2.38)

and we denote byZ₁^p the associatedmark, uniquely defined byN({S₁^p} × {Z₁^p}) = 1.

We next define recursivelyO^p_i,τi,S_i^p, andZ_i^pfori∈ {1, ..., α}, in the following way. Leti∈ {1, ..., α−1} be fixed. Due to (H5)(x0), we may build, for any ψ1, ..., ψi+1 ∈ D^x0,2α,, any z1 ∈ O₁^p(ψ1), ..., zi ∈ O^p_i(ψ1, ..., ψi;z1, ..., zi−1), an increasing sequence {O^p_i+1(ψ1, ..., ψi+1;z1, ..., zi)}^p≥1 of subsets ofRⁿ such that

∪^p≥1O^p_i+1(ψ1, ..., ψi+1;z1, ..., zi) =Oi+1(ψ1, ..., ψi+1;z1, ..., zi), and∀p≥1,

Z

O^p_i+1(ψ1,...,ψi+1;z1,...,zi)

ϕ(z)dz=p. (2.39)

We also consider the stopping times

τi+1 = infn

s > S_i^p; ψS^p_i,s∈ D/ ^x0,2α,

o

, (2.40)

S_i+1^p = inf (

s > S_i^p; Z s

S^p_i

Z

Rⁿ

1_{z∈O^p

i+1(ψ_0,Sp 1−,ψ_Sp

1,Sp 2−,...,ψ_Sp

i,(u∧τi+1 )−;Z₁^p,...,Z_i^p)}N(du, dz)≥1 )

,

and we denote byZ_i+1^p the associatedmark, uniquely defined byN({S_i+1^p } × {Z_i+1^p }) = 1.

Step 2: We remark that

for Ω⁰_p=∩i∈{1,...,α}{τi≥S_i^p}, lim

p→∞P[Ω⁰_p] = 1, (2.41) and for Ω¹_p= Ω⁰_p∩ {S_α^p ≤t}, lim

p→∞P[Ω¹_p] = 1. (2.42) Noting that (S_i+1^p −S^p_i)i∈{1,...,α} is a family of independent exponentially distributed random variables with parameter p, (2.41) follows from Lemma 2.11 and the strong Markov property. Using finally that t >0 while S_α^p follows a Gamma distribution with parametersαandpallows to obtain (2.42).

Step 3: We now show that the law ofX_S^x0α

p conditionally toS_p^αhas a density with respect to the Lebesgue measure onR^don the set Ω⁰_p. First note that on Ω⁰_p,

X_S^x0p

α=g

x0,ψ_0,Sp 1−,...,ψ_Sp

α−1,Sp α−

α (Z₁^p, ..., Z_α^p). (2.43)

Indeed, recalling Notation 2.8, one can check thatX_S^x0p

1−=ψ0,S^p₁−(x0), so that X_S^xp0

1 =ψ0,S^p₁−(x0) +hh

ψ0,S^p₁−(x0), Z₁^pi

=g₁^{x0,ψ0,Sp1−}(Z₁^p). (2.44)

ThusX_S^x0p

2−=ψ_S1^p_,S2^p₋

g^x₁^0,ψ0,S1p−(Z₁^p)

, so that

X_S^x0p

2 =ψS^p₁,S^p₂−

g₁^{x0,ψ0,Sp1−}(Z₁^p)

+h

ψS₁^p,S₂^p−

g^x₁^0,ψ0,S1p−(Z₁^p)

, Z₂^p

=g₂^{x0,ψ0,Sp1−,ψSp1,S2p−}(Z₁^p, Z₂^p), (2.45) and so on...

Consider theσ-field

G^p=σn

ψ_0,S^p₁₋, ..., ψ_S^p

α−1,Sα^p−, S₁^p, ..., S_α^po

. (2.46)

We next claim that our construction leads to the conclusion that on Ω^p₀, the law of (Z₁^p, ..., Z_α^p) condi- tionally toG^p is given by

1

p1_{z1∈O^p

1(ψ_0,Sp

1−)}ϕ(z1)dz1×1

p1_{z2∈O^p

2(ψ_0,Sp 1−,ψ_Sp

1,Sp

2−,z1)}ϕ(z2)dz2

×...×1

p1_{zα∈O^p_α(ψ

0,Sp 1−,...,ψ_Sp

α−1,Sp

α−;z1,...,zα−1)}ϕ(zα)dzα. (2.47) Hence, for any Lebesgue-null setA∈ A, since Ω^p₀ isG^p-measurable,

P[Ω⁰_p, X_S^x0p

α ∈A| G^p] =1_Ω0 pP

g

x0,ψ_0,Sp 1−,...,ψ_Sp

α−1,Sp α−

α (Z₁^p, ..., Z_α^p)∈A G^p

=1_Ω⁰

p

1 p^α

Z

O^p₁(ψ_0,Sp 1−)

ϕ(z1)dz1...

Z

Oα^p(ψ_0,Sp 1−,...,ψ_Sp

α−1,Sp

α−;z1,...,zα−1)}

ϕ(zα)dzα

1⁽

g

x0,ψ 0,Sp

1−,...,ψ Sp

α−1,Sp α−

α (z1,...,zα)∈A

)= 0 a.s., (2.48)

where the last equality comes from (H5)(x0) (see (2.28)). SinceS^α_p isG^p-measurable, this concludes the Step.