Chapter 1 Integration by parts formula with respect to jump times for stochastic differential equations

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1 Chapter Integration by parts formula with respect to jump times for stochastic differential equations Vlad Bally and Emmanuelle Clément Abstract We establish an integration by parts formula based on jump times in an abstract framewor in order to study the regularity of the law for processes solution of stochastic differential equations with jumps. Key words: Integration by parts formula, Stochastic Equations, Poisson Point Measures. 200 MSC: Primary: 60H07, Secondary: 60G55, 60G57. Introduction We consider the one-dimensional equation X t = x + 0 E c(u, a, X u )dn(u, a) + 0 g(u, X u )du (.) where N is a Poisson point measure of intensity measure µ on some abstract measurable space E. We assume that c and g are infinitely differentiable with respect to t and x, have bounded derivatives of any order and have linear growth with respect to x. Moreover we assume that the derivatives of c are bounded by a function c such that c(a)dµ(a) <. Under these E Vlad Bally Université Paris-Est, Laboratoire d Analyse et de Mathématiques Appliquées, UMR 8050, 5 Bld Descartes, Champs-sur-marne, Marne-la-Vallée Cedex 2, France vlad.bally@univ-mlv.fr Emmanuelle Clément Université Paris-Est, Laboratoire d Analyse et de Mathématiques Appliquées, UMR 8050, 5 Bld Descartes, Champs-sur-marne, Marne-la-Vallée Cedex 2, France emmanuelle.clement@univ-mlv.fr

2 2 Vlad Bally and Emmanuelle Clément hypotheses the equation has a unique solution and the stochastic integral with respect to the Poisson point measure is a Stieltjes integral. Our aim is to give sufficient conditions in order to prove that the law of X t is 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 Bismut [4] and then in Bichteler, Gravereaux and Jacod [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 taing care of the regularity. A first result in this sense was obtained by Carlen and Pardoux [5] and was followed by a lot of other papers (see [6], [7], [], [3]). The second step is to obtain the regularity of the density. Recently two results of this type have been obtained by Ishiawa and Kunita [0] and by Kuli [2]. 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) where U is a Lévy 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 ind of difficulties in this paper). Ishiawa and Kunita [0] used the finite difference approach given by J. Picard [4] in order to obtain sufficient conditions for the regularity of the density of the solution of an equation of type (.) (in a somehow more particular form, close 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 Kuli [2]. 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 coefficient f 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 µ is singular. He also proves that under some appropriate conditions, the density is not smooth for a small t so 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 x c + t c)(t, a, x). Except the regularity and boundedness conditions on g and c we consider the following non degeneracy assumption. There exists a measurable function α

3 Integration by parts formula with respect to jump times for SDE s 3 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 subsets E n E such that µ(e n ) < and lim n µ(e n ) ln( dµ(a)) = θ <. E n α(a) We need this hypothesis in order to control the error due to the fact that we localize our differential calculus on a non degeneracy set. If θ = 0 then, for every t > 0, the law of X t has a C density with respect to the Lebesgue measure. Suppose now that θ > 0 and let q N. Then, for t > 6θ(q + 2)(q + ) 2 the law of X t has a density of class C q. Notice that for small t we are not able to prove that a density exists and we have to wait for a sufficiently large t in order to obtain a regularization effect. In the paper of Kuli [2] one taes 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 Ishiawa and Kunita the basic example (which corresponds to the geometric Lévy process) is c(t, a, x) = xa(e a ) and g constant. So α(t, a, x) = a(e a ) a 2 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 []. But there are two specific points: first of all the integration by parts formula tae into account the border terms (in the above mentioned papers the border terms cancel because one maes use of some weights which are null on the border; but in the paper of Kuli [2] 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.

4 4 Vlad Bally and Emmanuelle Clément.2 Integration by parts formula.2. Notations-derivative operators The abstract framewor is quite similar to the one developed in Bally and Clément [2] but we introduce here some modifications in order to tae 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 variable J, G measurable, with values in N. Our aim is to establish a differential calculus based on the variables (V i ), conditionally on G. 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 (V i ) i J are independent and for each i {,..., 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 {,..., J}, there exist some G measurable random variables a i and b i such that < a i < b i < +, (a i, b i ) {p i > 0}. We also assume that p i admits a continuous bounded derivative on (a i, b i ) and that ln p i is bounded on (a i, b i ). We define now the class of functions on which this differential calculus will apply. We consider in this paper functions f : Ω R N R which can be written as f(ω, v) = f m (ω, v,..., v m ) {J(ω)=m} (.3) m= where f m : Ω R m R are G 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 l L, we fix a set of indices I l {,..., J} such that if l 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 and end with V i, i I. Moreover, given l we fix a partition (Λ l,i ) i Il of Ω such that the sets Λ l,i G, i I l. If ω 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 variables F = f(ω, V ) where V = (V i ) i and f is given by (.3). To simplify the notation we write F = f J (ω, V,..., V J ) so that conditionally on G we have J = m and F = f m (ω, V,..., V m ). We denote by S 0 the space of random variables F = f J (ω, V,..., V J ) where f J is a continuous function on O J = J i= (a i, b i ) such that there exists a G measurable random variable C satisfying sup f J (ω, v) C(ω) < + a.e. (.4) v O J

5 Integration by parts formula with respect to jump times for SDE s 5 We also assume that f J has left limits (respectively right limits) in a i (respectively in b i ). Let us be more precise. With the notations V (i) = (V,..., V i, V i+,..., V J ), (V (i), v i ) = (V,..., V i, v i, V i+,..., V J ), for v i (a i, b i ) our assumption is that the following limits exist and are finite: lim f J (ω, V (i), a i + ε) := F (a + ε 0 i ), lim f J (ω, V (i), b i ε) := F (b ε 0 i ). (.5) Now for, S (I l ) denotes the space of random variables F = f J (ω, V,..., V J ) S 0, such that f J admits partial derivatives up to order with respect to the variables v i, i I l and these partial derivatives belong to S 0. We are now able to define our differential operators. The derivative operators. We define D l : S (I l ) S 0 (I l ) : by D l F := OJ (V ) i I l Λl,i (ω) vi f(ω, V ), where O J = J i= (a i, b i ). The divergence operators We note p (l) = i I l Λl,i p i, (.6) and we define δ l : S (I l ) S 0 (I l ) by δ l (F ) = D l F + F D l ln p (l) = OJ (V ) i I l Λl,i ( vi F + F vi ln p i ) We can easily see that if F, U S (I l ) we have δ l (F U) = F δ l (U) + UD l F. (.7) The border terms (.5) ) Let U S 0 (I l ). We define (using the notation [U] l = i I l Λl,i OJ,i (V (i) )((Up i )(b i ) (Up i)(a + i )) with O J,i = j J,j i (a j, b j )

6 6 Vlad Bally and Emmanuelle Clément.2.2 Duality and basic integration by parts formula In our framewor 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. Assuming H0 then F, U S (I l ) we have E G (UD l F ) = E G (F δ l (U)) + E G [F U] l. (.8) For simplicity, we assume in this proposition that the random variables F and U tae values in R but such a result can easily be extended to R d value random variables. Proof. We have E G (UD l F ) = i I l Λl,i E G OJ (V )( vi f J (ω, V )u J (ω, V )). From H0 we obtain E G OJ (V )( vi f J (ω, V )u J (ω, V )) = E G OJ,i (V (i) ) bi By using the classical integration by parts formula, we have a i vi (f J )u J p i (v i )dv i. bi a i bi vi (f J )u J p i (v i )dv i = [f J u J p i ] bi a i f J vi (u J p i )dv i. a i Observing that vi (u J p i ) = ( vi (u J ) + u J vi (ln p i ))p i, we have E G ( OJ (V ) vi f J u J ) = E G OJ,i [(V (i) )f J u J p i ] bi a i E G OJ (V )F ( vi (U) + U vi (ln p i )) and the proposition is proved. We can now state a first integration by parts formula. Proposition 2. Let H0 hold true and let F S 2 (I l ), G S (I l ) and Φ : R R be a bounded function with bounded derivative. We assume that F = f J (ω, V ) satisfies the condition min i I l where γ is G measurable and we define on {γ > 0} then inf vi f J (ω, v) γ(ω), (.9) v O J (D l F ) = OJ (V ) i I l Λl,i vi f(ω, V ), {γ>0} E G (Φ () (F )G) = {γ>0} E G (Φ(F )H l (F, G)) + {γ>0} E G [Φ(F )G(D l F ) ] l (.0)

7 Integration by parts formula with respect to jump times for SDE s 7 with H l (F, G) = δ l (G(D l F ) ) = Gδ l ((D l F ) ) + D l G(D l F ). (.) Proof. We observe that D l Φ(F ) = OJ (V ) Λl,i vi Φ(F ) = OJ (V )Φ () (F ) Λl,i vi F, i I l i I l so that D l Φ(F ).D l F = Φ () (F )(D l F ) 2, and then {γ>0} Φ () (F ) = {γ>0} D l Φ(F ).(D l F ). Now since F S 2 (I l ), we deduce that (D l F ) S (I l ) on {γ > 0} and applying Proposition with U = G(D l F ) 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 I for l. In order to give some estimates of the weights appearing in these formulas we introduce the following norm on S l ( l = I ), for l L. F l = F + where. is defined on S 0 by l = l <...<l l F = sup v O J f J (ω, v). D l... D l F, (.2) For l = 0, we set F 0 = F. We remar that we have for l <... < l l D l... D l F = i I l,...,i I l ( Λlj,i j ) vi... vi F, and since for each l (Λ l,i ) i Il 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+ ( l+ = I ) : j= F l+ = D l+ F l + F l so D l+ F l F l+. (.3)

8 8 Vlad Bally and Emmanuelle Clément Moreover it is easy to chec that if F, G S l ( l = I ) F G l C l F l G l, (.4) where C l is a constant depending on l. Finally for any function φ C l (R, R) we have φ(f ) l C l l =0 φ () (F ) F l C l max 0 l φ() (F ) ( + F l l). (.5) With these notations we can iterate the integration by parts formula. Theorem. Let H0 hold true and let Φ : R R be a bounded function with bounded derivatives up to order L. Let F = f J (w, V ) S ( L l= I l) such that inf i {,...,J} inf vi f J (ω, v) γ(ω), γ [0, ] Gmeasurable (.6) v O J then we have for l {,..., L}, G S l ( l = I ) and F S l+ ( l = I ) {γ>0} E G Φ (l) (F )G C l Φ {γ>0} E G ( G l ( + p 0 ) l Π l (F ) ) (.7) where Φ = sup x Φ(x), p 0 = max l=,...,l p (l), C l is a constant depending on l and Π l (F ) is defined on {γ > 0} by Π l (F ) = l ( + (D F ) )( + δ ((D F ) ) ). (.8) = Moreover we have the bound Π l (F ) C l ( + ln p ) l γ l(l+2) l ( + F = where ln p = max i=,...,j (ln p i ). + D F ) 2, (.9) Proof. We proceed by induction. For l =, we have from Proposition 2 since G S (I ) and F S 2 (I ) {γ>0} E G (Φ () (F )G) = {γ>0} E G (Φ(F )H (F, G)) We have on {γ > 0} + {γ>0} E G [Φ(F )G(D F ) ]. (.20) H (F, G) G δ ((D F ) ) + D G (D F ), ( G + D G )( + (D F ) )( + δ ((D F ) ) ), = G ( + (D F ) 0 )( + δ ((D F ) ) 0 ). Turning to the border term [Φ(F )G(D F ) ], we chec that

9 Integration by parts formula with respect to jump times for SDE s 9 [Φ(F )G(D F ) ] 2 Φ G i I Λ,i vi F i I Λ,i p i, 2 Φ G 0 (D F ) 0 p 0. This proves the result for l =. Now assume that Theorem is true for l and let us prove it for l+. By assumption we have G S l+ ( l+ = I ) S (I l+ ) and F S l+2 ( l+ = I ) S 2 (I l+ ). Consequently we can apply Proposition 2 on I l+. This gives ) {γ>0} E G (Φ (l+) (F )G) = {γ>0} E G (Φ (l) (F )H l+ (F, G) + {γ>0} E G [Φ (l) (F )G(D l+ F ) ] l+, (.2) with H l+ (F, G) = Gδ l+ ((D l+ F ) ) + D l+ G(D l+ F ), [Φ (l) (F )G(D l+ F ) ] l+ = ( Λl+,i OJ,i (V (i) ) (Φ (l) (F )G vi F p i)(b i ) i I l+ (Φ (l) (F )G ) vi F p i)(a + i ). We easily see that H l+ (F, G) S l ( l = I ) and so using the induction hypothesis we obtain {γ>0} E G Φ (l) (F )H l+ (F, G) C l Φ {γ>0} E G H l+ (F, G) l (+ p 0 ) l Π l (F ), and we just have to bound H l+ (F, G) l on {γ > 0}. But using successively (.4) and (.3) H l+ (F, G) l C l ( G l δ l+ ((D l+ F ) ) l + D l+ G l (D l+ F ) ) l, C l G l+ ( + (D l+ F ) ) l )( + δ l+ ((D l+ F ) ) l ). This finally gives E G Φ (l) (F )H l+ (F, G) C l Φ E G G l+ ( + p 0 ) l Π l+ (F ). (.22) So we just have to prove a similar inequality for E G [Φ (l) (F )G(D l+ F ) ] l+. This reduces to consider i I l+ Λl+,i p i (b i )E G OJ,i (V (i) )(Φ (l) (F )G vi F )(b i ) (.23) since the other term can be treated similarly. Consequently we just have to bound E G OJ,i (V (i) )(Φ (l) (F )G vi F )(b i ).

10 0 Vlad Bally and Emmanuelle Clément Since all variables satisfy (.4), we obtain from Lebesgue Theorem, using the notation (.5) E G OJ,i (V (i) )(Φ (l) (F )G vi F )(b i ) = lim E G OJ,i (V (i) )(Φ (l) (f J (V (i), b i ε))(g J ε 0 vi f J )(V (i), b i ε). To shorten the notation we write simply F (b i ε) = f J (V (i), b i ε). Now one can prove that if U S l ( l+ = I ) for l l then i I l+, U(b i ε) S l ( l = I ) and U(b i ε) l U l. We deduce then that i I l+ F (b i ε) S l+ ( l = I ) and that (G vi F )(b i ε) S l ( l = I ) and from induction hypothesis E G (Φ (l) (F (b i ε)) OJ,i (G vi F )(b i ε) C l Φ E G { G(b i ε) l vi F (b i ε) l ( + p 0 ) l Π l (F (b i ε))}, C l Φ E G G l vi F l( + p 0 ) l Π l (F ). Putting this in (.23) we obtain E G Λl+,i OJ,i (Φ (l) (F )G vi F p i)(b i ) C l Φ E G { G l ( + p 0 ) l Π l (F ) i I l+ Finally plugging (.22) and (.24) in (.2) Λl+,i p i vi F l}, i I l+ C l Φ E G { G l ( + p 0 ) l+ Π l (F ) (D l+ F ) l }. (.24) E G (Φ (l+) (F )G) C l Φ ( EG G l+ ( + p 0 ) l Π l+ (F ) +E G G l ( + p 0 ) l+ Π l (F ) (D l+ F ) l ), C l Φ E G G l+ ( + p 0 ) l+ Π l+ (F ), and inequality (.7) is proved for l +. This achieves the first part of the proof of Theorem. It remains to prove (.9). We assume that ω {γ > 0}. Let l. We first notice that combining (.3) and (.4) we obtain δ (F ) F ( + D ln p () ), since p () only depends on the variables V i, i I. So we deduce the bound Now we have δ ((D F ) ) (D F ) ( + ln p ). (.25)

11 Integration by parts formula with respect to jump times for SDE s From (.5) with φ(x) = /x and consequently (D F ) = Λ,i vi F i I vi F ( + F ) C γ, (D F ) C ( + F ) γ. (.26) Moreover we have using successively (.3) and (.26) Putting this in (.25) (D F ) = (D ( F ) + D (D F ), (+ F ) ) C + (+ D F γ ), γ + (+ F C + D F ) γ +. δ ((D F ) ) ( + F + D F ) C γ + ( + ln p ). (.27) Finally from (.26) and (.27), we deduce Π l (F ) C l ( + ln p ) l γ l(l+2) l ( + F = + D F ) 2, and Theorem is proved..3 Stochastic equations with jumps.3. Notations and hypotheses We consider a Poisson point process p with measurable state space (E, B(E)). We refer to Ieda and Watanabe [9] for the notation. We denote by N the counting measure associated to p so N t (A) := N((0, t) A) = #{s < t; p s A}. The intensity measure is dt dµ(a) where µ is a sigma-finite measure on (E, B(E)) and we fix a non decreasing sequence (E n ) of subsets of E such that E = n E n, µ(e n ) < and µ(e n+ ) µ(e n ) + K for all n and for a constant K > 0. We consider the one dimensional stochastic equation

12 2 Vlad Bally and Emmanuelle Clément X t = x + c(s, a, X s )dn(s, a) + g(s, X s )ds. (.28) 0 E 0 Our aim is to give sufficient conditions on the coefficients c and g in order to prove that the law of X t is absolutely continuous with respect to the Lebesgue measure and has a smooth density. We mae the following assumptions on the coefficients c and g. H. We assume that the functions c and g are infinitely differentiable with respect to the variables (t, x) and that there exist a bounded function c and a constant g, such that (t, a, x) c(t, a, x) c(a)( + x ), sup t l xc(t, l a, x) c(a); l+l (t, x) g(t, x) g( + x ), sup t l xg(t, l x) g; l+l We assume moreover that c(a)dµ(a) <. E Under H, equation (.28) admits a unique solution. H2. We assume that there exists a measurable function ĉ : E R + such that ĉ(a)dµ(a) < and E (t, a, x) x c(t, a, x)( + x c(t, a, x)) ĉ(a). To simplify the notation we tae ĉ = c. Under H2, the tangent flow associated to (.28) is invertible. At last we give a non-degeneracy condition wich will imply (.6). We denote by α the function α(t, a, x) = g(t, x) g(t, x + c(t, a, x)) + (g x c + t c)(t, a, x). (.29) H3. We assume that there exists a measurable function α : E R + such that (t, a, x) α(t, a, x) α(a) > 0, n dµ(a) < and lim inf E n α(a) n µ(e n ) ln ( E n α(a) dµ(a) ) = θ <. We give in the following some examples where E = (0, ] and α(a) = a..3.2 Main results and examples Following the methodology introduced in Bally and Clément [2], our aim is to bound the Fourier transform of X t, ˆp Xt (ξ), in terms of / ξ, recalling that if R ξ q ˆp Xt (ξ) dξ <, for q > 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

13 Integration by parts formula with respect to jump times for SDE s 3 proposition relies on an approximation of X t which will be given in the next section. Proposition 3. Assuming H, H2 and H3 we have for all n, L N ( ˆp Xt (ξ) C t,l e µ(en)t/(2l) + ) ξ L A n,l, with A n,l = µ(e n ) L ( E n α(a) dµ(a))l(l+2). From this proposition, we deduce our main result. Theorem 2. We assume that H, H2 and H3 hold. Let q N, then for t > 6θ(q + 2)(q + ) 2, the law of X t is absolutely continuous with respect to the Lebesgue measure and its density is of class C q. In particular if θ = 0, the law of X t is absolutely continuous with respect to the Lebesgue measure and its density is of class C for every t > 0. Proof. From Proposition 3, we have ( ˆp Xt (ξ) C t,l e µ(en)t/2l + ) ξ L A n,l. Now, 0 > 0, if t/2l > θ, we deduce from H3 that for n n L t/2l > µ(e n ) ln( E n α(a) dµ(a)) + ln µ(e n) 0 µ(e n ) since the second term on the right hand side tends to zero. This implies e µ(en)t/2l > ( E n α(a) dµ(a)) µ(e n ) /0. Choosing = L(L + 2) and / 0 = L, we obtain that for n n L and 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 + e µ(en)t/2l, ξ L C t,l ( B + Bn(t) n(t) ), ξ L with B n (t) = e µ(en)t/2l. Now recalling that µ(e n ) < µ(e n+ ) K + µ(e n ), we have B n (t) < B n+ (t) K t B n (t). Moreover since B n (t) goes to infinity with n we have { ξ L/2 B nl (t)} = n n L {Bn(t) ξ L/2 <B n+(t)}. But if B n (t) ξ L/2 < B n+ (t), ˆp Xt (ξ) C t,l / ξ L/2 and so

14 4 Vlad Bally and Emmanuelle Clément ξ q ˆp Xt (ξ) dξ = ξ L/2 <B nl(t) ξ q ˆp Xt (ξ) dξ + ξ L/2 B nl (t) ξ q ˆp Xt (ξ) dξ, C t,l,nl + ξ L/2 B nl (t) ξ q L/2 dξ. For q N, choosing L such that L/2 q >, we obtain 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 > 2L 2 (L + 2)θ and L > 2(q + ), that is t > 6θ(q + ) 2 (q + 2) and Theorem 2 is proved. We end this section with two examples Example. We tae E = (0, ], µ λ = δ λ / with 0 < λ < and E n = [/n, ]. We have n E n = E, µ(e n ) = n = and µ λ λ (E n+ ) µ λ (E n )+. We consider the process (X t ) solution of (.28) with c(t, a, x) = a and g(t, x) = g(x) assuming that the derivatives of g are bounded and that g (x) g > 0. We have E adµ λ(a) = < so H and H2 hold. λ+ Moreover α(t, a, x) = g(x) g(x + a) so α(a) = ga. Now E n a dµ λ(a) = n = λ which is equivalent as n go to infinity to n 2 λ /(2 λ). Now we have µ λ (E n ) ln ( E n α(a) dµ λ(a) ) = ln(g n = λ ) n = n C ln(n2 λ ) n λ 0, λ and then H3 is satisfied with θ = 0. We conclude from Theorem 2 that t > 0, X t admits a density C. In the case λ =, we have µ (E n ) = n = n ln n then ( ) µ (E n ) ln E n α(a) dµ (a) = ln(g n = ) n, n = and this gives H3 with θ =. So the density of X t is regular as soon as t is large enough. In fact it is proved in Kuli [2] that under some appropriate conditions the density of X t is not continuous for small t. Example 2. We tae the intensity measure µ λ as in the previous example and we consider the process (X t ) solution of (.28) with g = and c(t, a, x) = ax. This gives c(a) = a and α(a) = a. So the conclusions are similar to example in both cases 0 < λ < and λ =. But in this example we can compare our result to the one given by Ichiawa and Kunita [0]. They assume the condition lim inf u 0 for some h (0, 2). Here we have a u u h a 2 dµ(a) > 0, a u a 2 dµ(a) = /u 2+λ u 0 ( ) u +λ + λ.

15 Integration by parts formula with respect to jump times for SDE s 5 So if 0 < λ <, ( ) holds and their results apply. In the case λ =, ( ) fails and they do not conclude. However in our approach we conclude that the density of X t is C q for t > 6(q + 2)(q + ) 2. The next section is devoted to the proof of Proposition Approximation of X t and integration by parts formula In order to bound the Fourier transform of the process X t solution of (.28), we will apply the differential calculus developed in section 2. The first step consists in an approximation of X t by a random variable Xt N which can be viewed as an element of our basic space S 0. We assume that the process (Xt N ) is solution of the discrete version of equation (.28) X N t = x + 0 c(s, a, Xs N )dn(s, a) + E N 0 g(s, X N s )ds. (.30) Since µ(e N ) <, the number of jumps of the process X N on the interval (0, t) is finite and consequently we may consider the random variable Xt N as a function of these jump times and apply the methodology proposed in section 2. We denote by (Jt N ) the Poisson process defined by Jt N = N((0, t), E N ) = #{s < t; p s E N } and we note (T N) its jump times. We also introduce the notation N = p T N. With these notations, the process solution of (.30) can be written Jt N Xt N = x + c(t N, N, X N T N ) + g(s, Xs N )ds. (.3) = We will not wor with all the variables (T N) but only with the jump times (T n) of the Poisson process J t n, where n < N. In the following we will eep n fixed and we will mae N go to infinity. We note (T N,n ) the jump times of the Poisson process J N,n t = N((0, t), E N \E n ) and n,n = p T n,n. Now we fix L N, the number of integration by parts and we note t l = tl/l, 0 l L. Assuming that Jt n l Jt n l = m l for l L, we denote by (Tl,i n ) i m l the jump times of Jt n belonging to the time interval (t l, t l ). In the following we assume that m l, l. For i = 0 we set Tl,0 n = t l and for i = m l +, Tl,m n l + = t l. With these definitions we choose our basic variables (V i, i I l ) as (V i, i I l ) = (Tl,2i+, n 0 i [(m l )/2]). (.32) The σ-algebra which contains the noise which is not involved in our differential calculus is 0

16 6 Vlad Bally and Emmanuelle Clément G = σ{(j n t l ) l L ; (T n l,2i) 2i ml, l L; (T N,n ) ; ( N ) }. (.33) Using some well nown results on Poisson processes, we easily see that conditionally on G the variables (V i ) are independent and for i I l the law of V i conditionally on G is uniform on (Tl,2i n, T l,2i+2 n ) and we have p i (v) = Tl,2i+2 n T l,2i n (T n l,2i,tl,2i+2 n ) (v), i I l, (.34) Consequently taing a i = Tl,2i n and b i = Tl,2i+2 n we chec that hypothesis H0 holds. It remains to define the localizing sets (Λ l,i ) i Il. We denote h n l = t l t l = t 2m l 2Lm l and n l = [(m l )/2]. We will wor on the G measurable set Λ n l = n l i=0 {T n l,2i+2 T n l,2i h n l }, (.35) and we consider the following partition of this set: Λ l,0 = {T n l,2 T n l,0 h n l }, Λ l,i = i ={T n l,2 T n l,2 2 < h n l } {T n l,2i+2 T n l,2i h n l }, i =,..., n l. After L l iterations of the integration by parts we will wor with the variables V i, i I l so the corresponding derivative is D l F = i I l Λl,i Vi F = i I l Λl,i T n l,2i+ F. If we are on Λ n l then we have at least one i such that t l Tl,2i n < Tl,2i+ n < T l,2i+2 n t l and Tl,2i+2 n T l,2i n h n l. Notice that in this case Λl,i p i (h n l ) and roughly speaing this means that the variable V i = Tl,2i+ n gives a sufficiently large quantity of noise. Moreover, in order to perform L integrations by parts we will wor on Γ n L = L l=λ n l (.36) and we will leave out the complementary of ΓL n. The following lemma says that on the set ΓL n we have enough noise and that the complementary of this set may be ignored. Lemma. Using the notation given in Theorem one has i) p 0 := max l L i I l Λl,i p i 2L t J n t, ii) P ((Γ n L )c ) L exp( µ(e n )t/2l). Proof. As mentioned before Λl,i p i (h n l ) = 2Lm l /t 2L t J t n and so we have i). In order to prove ii) we have to estimate P ((Λ n l )c ) for l L.

17 Integration by parts formula with respect to jump times for SDE s 7 We denote s l = 2 (t l + t l ) and we will prove that {Jt n l Js n l } Λ n l. Suppose first that m l = Jt n l Jt n l is even. Then 2n l + 2 = m l. If Tl,2i+2 n T l,2i n < hn l for every i = 0,..., n l then T n l,m l t l = n l i=0 (T n l,2i+2 T n l,2i) (n l + ) t t 2Lm l 4L s l t l so there are no jumps in (s l, t l ). Suppose now that m l is odd so 2n l +2 = m l + and Tl,2n n l +2 = t l. If we have Tl,2i+2 n T l,2i n < hn l for every i = 0,..., n l, then we deduce n l i=0 (T n l,2i+2 T n l,2i) < (n l + ) t < m l + t 2Lm l 4L m l t 2L, and there are no jumps in (s l, t l ). So we have proved that {Jt n l Js n l } Λ n l and since P (Jt n l Js n l = 0) = exp( µ(e n )t/2l) the inequality ii) follows. Now we will apply Theorem, with F N = Xt N, G = and Φ ξ (x) = e iξx. So we have to chec that F N S L+ ( L l= I l) and that condition (.6) holds. Moreover we have to bound F N l l and D l F N l l, for l L. This needs some preliminary lemma. Lemma 2. Let v = (v i ) i 0 be a positive non increasing sequence with v 0 = 0 and (a i ) i a sequence of E. We define J t (v) by J t (v) = v i if v i t < v i+ and we consider the process solution of J t X t = x + c(v, a, X v ) + = 0 g(s, X s )ds. (.37) We assume that H holds. Then X t admits some derivatives with respect to v i and if we note U i (t) = vi X t and W i (t) = v 2 i X t, the processes (U i (t)) t vi and (W i (t)) t vi solve respectively J t U i (t) = α(v i, a i, X vi )+ x c(v, a, X v )U i (v )+ x g(s, X s )U i (s)ds, =i+ v i (.38) J t W i (t) = β i (t) + x c(v, a, X v )W i (v ) + x g(s, X s )W i (s)ds, v i with =i+ α(t, a, x) = g(t, x) g(t, x + c(t, a, x)) + g(t, x) x c(t, a, x) + t c(t, a, x), (.39) β i (t) = t α(v i, a i, X vi ) + x α(v i, a i, X vi )g(v i, X vi ) x g(v i, X vi )U i (v i ) + J t =i+ 2 xc(v, a, X v )(U i (v )) 2 + v i xg(s, 2 X s )(U i (s)) 2 ds.

18 8 Vlad Bally and Emmanuelle Clément Proof. If s < v i, we have vi X s = 0. Now we have and consequently v i vi X vi = x + c(v, a, X v ) + g(s, X s )ds, 0 For t > v i, we observe that this gives X t = X vi + = vi X vi = g(v i, X vi ). J t =v i c(v, a, X v ) + v i g(s, X s )ds, vi X t = g(v i, X vi ) + g(v i, X vi ) x c(v i, a i, X vi ) + t c(v i, a i, X vi ) g(v i, X vi ) + J t =i+ xc(v, a, X v ) vi X v + v i x g(s, X s ) vi X s ds. Remaring that X vi = X vi + c(v i, a i, X vi ), we obtain (.38). The proof of (.39) is similar and we omit it. We give next a bound for X t and its derivatives with respect to the variables (v i ). Lemma 3. Let (X t ) be the process solution of (.37). We assume that H holds and we note J t n t (c) = c(a ). Then we have: = sup X t C t ( + n t (c))e nt(c). s t Moreover l, there exist some constants C t,l and C l such that (v i ) i=,...,l with t > v l, we have sup v... vl U l (s) + sup v... vl W l (s) v l s t v l s t C t,l ( + n t (c)) C l e C ln t(c). We observe that the previous bound does not depend on the variables (v i ). Proof. We just give a setch of the proof. We first remar that the process (e t ) solution of J t e t = + c(a )e v + g e s ds, = 0

19 Integration by parts formula with respect to jump times for SDE s 9 is given by e t = J t = ( + c(a ))e gt. Now from H, we deduce for s t X s x + J s = c(a )( + X v ) + s 0 g( + X u )du, x + J t = c(a ) + gt + J s = c(a ) X v + s ( x + J t = c(a ) + gt)e s 0 g X u du, where the last inequality follows from Gronwall lemma. Then using the previous remar sup s t J t X s C t ( + n t (c)) ( + c(a )) C t ( + n t (c))e nt(c). (.40) = We chec easily that α(t, a, x) C( + x )c(a), and we get successively from (.38) and (.40) sup U l (s) C t (+ X vl )c(a l )(+n t (c))e nt(c) C t (+n t (c)) 2 e 2nt(c). v l s t Putting this in (.39), we obtain a similar bound for sup vl s t W l (s) and we end the proof of Lemma 3 by induction since we can derive equations for the higher order derivatives of U l (s) and W l (s) analogous to (.39). We come bac to the process (Xt N ) solution of (.30). We recall that F N = Xt N and we will chec that F N satisfies the hypotheses of Theorem. Lemma 4. i) We assume that H holds. Then l, C t,l, C l independent of N such that F N l + D l F N l C t,l (( + N t (c))e Nt(c)) C l, with N t (c) = c(a)dn(s, a). 0 E ii) Moreover if we assume in addition that H2 and H3 hold and that m l = Jt n l Jt n l, l {,..., L} then we have l L, i I l and (.6) holds. Vi F N ( e 2Nt(c) N t ( En /α)) := γn We remar that on the non degeneracy set ΓL n given by (.36) we have at least one jump on (t l, t l ), that is m l, l {,..., L}. Moreover we have ΓL n {γ n > 0}. Proof. The proof of i) is a straightforward consequence of Lemma 3, replacing n t (c) by J N t p= c( N p ) and observing that

20 20 Vlad Bally and Emmanuelle Clément Jt N c( N p ) = p= 0 E N c(a)dn(s, a) Turning to ii) we have from Lemma 2 N Jt T N Xt N = α(t N, N, X N T N ) + p=+ 0 E c(a)dn(s, a) = N t (c). x c(t N p, N p, X N T N p ) T N XN T N p + T N x g(s, X N s ) T N X s ds. Assuming H2, we define (Y N t ) t and (Z N t ) t as the solutions of the equations Y N t Z N t = + J N t p= xc(tp N, N p, X N Tp N )Y T N + 0 xg(s, Xs N )Ys N = J N xc(t N p, N p,xn T t p N ) p= + xc(tp N, N p,xn Tp N )Z T N 0 xg(s, Xs N )Zs N ds. ds, We have Y N t Z N t =, t 0 and Yt N e tg e Nt( E c) N e Nt(c), Zt N = e Nt(c). Now one can easily chec that Y N t T N Xt N = α(t N, N, X N T N )Y t N Z N T, N and using H3 and the preceding bound it yields T N X N t e 2Nt(c) α( N ). Recalling that we do not consider the derivatives with respect to all the variables (T N) but only with respect to (V i) = (Tl,2i+ n ) l,i with n < N fixed, we have l L and i I l J n t Vi Xt N e 2Nt(c) ( α( n = e 2Nt(c) N t ( En /α)), p ) and Lemma 4 is proved. p= With this lemma we are at last able to prove Proposition 3. Proof. From Theorem we have since Γ n L {γ n > 0} Γ n L E G Φ (L) (F N ) C L Φ Γ n L E G ( + p 0 ) L Π L (F N ).

21 Integration by parts formula with respect to jump times for SDE s 2 Now from Lemma i) we have p 0 2LJ n t /t and moreover we can chec that ln p = 0. So we deduce from Lemma 4 This finally gives Π L (F N ) C ( t,l ( + Nt (c))e Nt(c)) C L γn L(L+2) C t,l N t ( En /α) ( L(L+2) ( + N t (c))e Nt(c)) C L. E Γ n L Φ (L) (F N ) Φ C t,l E ( (J N t ) L N t ( En /α) L(L+2) (( + N t (c))e Nt(c)) C L ). (.4) Now we now from a classical computation (see for example [2]) that the Laplace transform of N t (f) satisfies Ee snt(f) = e tα f (s), α f (s) = ( e sf(a) )dµ(a). (.42) From H, we have c(a)dµ(a) <, so we deduce using (.42) with f = c E that, q > 0 ( E ( + N t (c))e Nt(c)) q Ct,q <. Since J n t is a Poisson process with intensity tµ(e n ), we have q > 0 E E(J n t ) q C t,q µ(e n ) q. Finally, using once again (.42) with f = En /α we see easily that q > 0 ( ) q EN t ( En /α) q C t,q E n α(a) dµ(a). Turning bac to (.4) and combining Cauchy-Schwarz inequality and the previous bounds we deduce ( ) L(L+2) E Γ n L Φ (L) (F N ) Φ C t,l µ(e n ) L E n α(a) dµ(a) = Φ C t,l A n,l. (.43) We are now ready to give a bound for ˆp X N t (ξ). We have ˆp X N t (ξ) = EΦ ξ (F N ), with Φ ξ (x) = e iξx. Since Φ (L) ξ (x) = (iξ) L Φ ξ (x), we can write ˆp X N t (ξ) = EΦ (L) ξ (F N ) / ξ L and consequently we deduce from (.43)

22 22 Vlad Bally and Emmanuelle Clément ˆp X N t (ξ) P ((ΓL n ) c ) + C t,l A n,l / ξ L. But from Lemma ii) we have P ((ΓL n ) c ) Le µ(en)t/(2l) and finally ˆp X N t (ξ) C L,t (e µ(en)t/(2l) + A n,l / ξ L). We achieve the proof of Proposition 3 by letting N go to infinity, eeping n fixed. References. Vlad Bally, Marie-Pierre Bavouzet, and Marouen Messaoud. Integration by parts formula for locally smooth laws and applications to sensitivity computations. Ann. Appl. Probab., 7():33 66, Vlad Bally and Emmanuelle Clément. Integration by parts formula and applications to equations with jumps. Preprint, Klaus Bichteler, Jean-Bernard Gravereaux, and Jean Jacod. Malliavin calculus for processes with jumps, volume 2 of Stochastics Monographs. Gordon and Breach Science Publishers, New Yor, Jean-Michel Bismut. Calcul des variations stochastique et processus de sauts. Z. Wahrsch. Verw. Gebiete, 63(2):47 235, Eric A. Carlen and Étienne Pardoux. Differential calculus and integration by parts on Poisson space. In Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 988), volume 59 of Math. Appl., pages Kluwer Acad. Publ., Dordrecht, L. Denis. A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Related Fields, 8(3): , Robert J. Elliott and Allanus H. Tsoi. Integration by parts for Poisson processes. J. Multivariate Anal., 44(2):79 90, Nicolas Fournier. Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump. Electron. J. Probab., 3:no. 6, 35 56, Nobuyui Ieda and Shinzo Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, Yasushi Ishiawa and Hiroshi Kunita. Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps. Stochastic Process. Appl., 6(2): , Alexey M. Kuli. Malliavin calculus for Lévy processes with arbitrary Lévy measures. Teor. Ĭmovīr. Mat. Stat., (72):67 83, Alexey M. Kuli. Stochastic calculus of variations for general Lévy processes and its applications to jump-type SDE s with non degenerated drift. Preprint, Ivan Nourdin and Thomas Simon. On the absolute continuity of Lévy processes with drift. Ann. Probab., 34(3):035 05, Jean Picard. On the existence of smooth densities for jump processes. Probab. Theory Related Fields, 05(4):48 5, 996.