Malliavin calculus for parabolic SPDEs with jumps

We study a parabolic SPDE driven by a white noise and a compensated Poisson measure. We rst de ne the solutions in a weak sense, and we prove the existence and the uniqueness of a weak solution. Then we use the Malliavin calculus in order to show that under some non-degeneracy assumptions, the law of the weak solution admits a density with respect to the Lebesgue measure. To this aim, we introduce two derivative operators associated with the white noise and the Poisson measure. The one associated with the Poisson measure is studied in detail. c © 2000 Elsevier Science B.V. All rights reserved. 0. Introduction Let T ¿ 0 be a positive time, let ( ;F; (Ft)t∈[0;T ]; P) be a probability space, and let L be a positive real number. We consider [0; T ]× [0; L] and [0; T ]× [0; L]×R endowed with their Borel elds. Let W (dx; dt) be a space–time white noise on [0; L]× [0; T ] based on dx dt (see e.g. Walsh, 1986, p. 269), and let N be a Poisson measure on [0; T ]× [0; L]×R, independent of W , with intensity measure (dt; dx; dz)=dt dxq(dz), where q is a positive nite measure on R . The compensated Poisson measure is denoted by Ñ=N− . Our purpose is to study the following one-dimensional stochastic partial di erential equation on [0; L]× [0; T ]: [ @V @t (x; t)− @ 2V @x2 (x; t) ] dx dt = g(V (x; t)) dx dt + f(V (x; t))W (dx; dt) + ∫ R h(V (x; t); z)Ñ (dt; dx; dz) (0.1) with Neumann boundary conditions ∀t ¿ 0; @V @x (0; t) = @V @x (L; t) = 0 (0.2) and with deterministic initial condition V (x; 0) =V0(x). E-mail address: [email protected] (N. Fournier) 0304-4149/00/$ see front matter c © 2000 Elsevier Science B.V. All rights reserved. PII: S0304 -4149(99)00107 -6 116 N. Fournier / Stochastic Processes and their Applications 87 (2000) 115–147 In this paper, we rst prove the existence and uniqueness of a weak solution {V (x; t)} to (0.1). Then we show, in the case where q(dz) admits a su ciently regular density, and under some non-degeneracy conditions, that the law of V (x; t) is absolutely continuous with respect to the Lebesgue measure as soon as t ¿ 0. Parabolic SPDEs driven by a white noise, i.e. Eq. (0.1) with h ≡ 0, have been introduced by Walsh (1981, 1986). Walsh (1986) de nes his weak solutions, then he proves a theorem of existence, uniqueness and regularity. Since then, various properties of Walsh’s equation have been investigated. In particular, the Malliavin calculus has been developed by Pardoux and Zhang (1993), and Bally and Pardoux (1998). But Walsh builds his equation in order to model a discontinuous neurophysiological phenomenon. Walsh (1981) explains that the white noise W approximates a Poisson point process. This approximation is realistic because there are many jumps, and the jumps are very small, but in any case, the observed phenomenon is discontinuous. However, SPDEs with jumps are much less known. In the case where f ≡ 0, Saint Loubert Bi e (1998) has studied the existence, uniqueness, regularity, and stochastic variational calculus. We prove here a result of existence and uniqueness, because we de ne in a slighlty di erent way the weak solutions, and also because Saint Loubert Bi e does not study exactly the same equation. Furthermore, his result about the absolute continuity does not extend to the present case. The Malliavin calculus for jump processes we will build extends the work of Bichteler et al. (1987), who study di usion processes with jumps. We cannot apply directly their methods, essentially because the weak solution of (0.1) is not a semi-martingale. Bichteler et al. (1987), use a “scalar product of derivation”, which does not allow to obtain satisfying results in the present case (see Saint Loubert Bi e, 1998). Thus we have to introduce a real “derivative operator”, which gives more information. Our method is also inspired by the paper of Bally and Pardoux (1998), who prove the existence of a smooth density in the case where h ≡ 0. The present work is organized as follows. In Section 1, we de ne the solutions of (0.1) in a weak sense, which is easy in the continuous case but slightly more di cult here, because there are “predictability” problems. Then we state our main results. An existence and uniqueness result is proved in Section 2. We study the existence of a density for the law of the weak solution in Section 3. Finally, an appendix is given at the end of the paper. 1. Statement of the main results In the whole work, we assume that ( ;F; {Ft}t∈[0;T ]; P) is the canonical product probability space associated with W and N . In particular, Ft = {W (A);A∈B([0; L]× [0; t])}∨ {N (B);B∈B([0; t]× [0; L]×R)}: (1.1) De nition 1.1. Consider a process Y = {Y (y; s)}[0;L]×[0;T ]. We will say that Y is • Predictable if it is Pred ⊗B([0; L])-measurable, where Pred is the predictable eld on × [0; T ]. N. Fournier / Stochastic Processes and their Applications 87 (2000) 115–147 117 • Bounded in L2 if sup[0;L]×[0;T ] E(Y 2(y; s))¡∞. • A version of X = {X (y; s)}[0;L]×[0;T ] if for all y∈ [0; L], all s∈ [0; T ], a.s., Y (y; s)= X (y; s). • A weak version of X={X(y; s)}[0;L]×[0;T ] if dP(!) dy ds-a.e., Y(y; s)(!)=X (y; s)(!). • Of class PV if it is bounded in L2 and if it is a weak version of a predictable process. We now de ne the stochastic integrals we will use. De nition 1.2. Let Y be a process that admits a predictable weak version Y−. Let be a measurable function such that ∫ T 0 ∫ L 0 ∫ R E( (Y (y; s); s; y; z))q(dz) dy ds¡∞: (1.2) Then we set ∫ T 0 ∫ L 0 ∫ R (Y (y; s); s; y; z)Ñ (ds; dy; dz) = ∫ T 0 ∫ L 0 ∫ R (Y−(y; s); s; y; z)Ñ (ds; dy; dz): (1.3) The obtained random variable does not depend on the choice of the predictable version, up to a P(d!)-negligible set. We de ne in the same way the stochastic integral against the white noise. Using the classical stochastic integration theory (see Jacod and Shiryaev, 1987, pp. 71–74; Walsh, 1986, pp. 292–298), we deduce, since Y− = Y dP dy ds-a.e., that E [(∫ T 0 ∫ L 0 ∫ R (Y (y; s); s; y; z)Ñ (ds; dy; dz) )2] = ∫ T 0 ∫ L 0 ∫ R E( (Y (y; s); s; y; z))q( dz) dy ds; (1.4) E [(∫ T 0 ∫ L 0 (Y (y; s); s; y)W ( dy; ds) )2 ] = ∫ T 0 ∫ L 0 E( (Y (y; s); s; y)) dy ds; (1.5) E [(∫ T 0 ∫ L 0 (Y (y; s); s; y) dy ds )2] 6TL ∫ T 0 ∫ L 0 E( (Y (y; s); s; y)) dy ds: (1.6) We would now like to de ne the weak solutions of (0.1). First, we suppose the following conditions, which in particular allow all the integrals below to be well de ned. Assumption (H). f and g satisfy some global Lipschitz conditions on R, h is measurable on R × R, and there exists a positive function ∈L2(R; q) such that for all x; y; z ∈R, |h(0; z)|6 (z) and |h(x; z)− h(y; z)|6 (z)|x − y|: (1.7) 118 N. Fournier / Stochastic Processes and their Applications 87 (2000) 115–147 Assumption (D). V0 is deterministic, B([0; L])-measurable, and bounded. Following Walsh (1981,1986), or Saint Loubert Bi e (1998), we de ne the weak solutions of (0.1) by using an evolution equation. Let Gt(x; y) be the Green kernel of the deterministic system: @u @t = @2u @x2 ; @u @x (0; t) = @u @x (L; t) = 0: (1.8) This means that Gt(x; y) is the solution of the system with initial condition a Dirac mass at y. It is well known that Gt(x; y)= 1 √ 4 t ∑ n∈Z [ exp (−(y − x − 2nL)2 4t ) +exp (−(y + x − 2nL)2 4t )] : (1.9) All the properties of G that we will use can be found in the appendix. Now we can de ne the weak solutions of Eq. (0.1). De nition 1.3. Assume (H) and (D). A process V of class PV is said to be a weak solution of (0.1) if for all x in [0; L], all t ¿ 0, a.s. V (x; t) = ∫ L 0 Gt(x; y)V0(y) dy + ∫ t 0 ∫ L 0 Gt−s(x; y)f(V (y; s))W (dy; ds) + ∫ t 0 ∫ L 0 Gt−s(x; y)g(V (y; s)) dy ds + ∫ t 0 ∫ L 0 ∫ R Gt−s(x; y)h(V (y; s); z)Ñ (ds; dy; dz); (1.10) where we have used De nition 1.2. Let us nally state our rst result. Theorem 1.4. Assume (H) and (D). Eq. (0:1) admits a unique solution V ∈PV in the sense of De nition 1:3. The uniqueness holds in the sense that if V ′ ∈PV is another weak solution; then V and V ′ are two versions of the same process, i.e. for each x; t; a.s.; V (x; t) = V ′(x; t). It is not standard to work with predictable weak versions. In the continuous case, no such problem appears, and the classical di usion processes with jumps are a.s. c adl ag. But here, the paths of a weak solution cannot be c adl ag in time. Indeed, this is even impossible in the much simpler case where V0 = 1, f = g = 0, h(x; z) = 1, where q(R)¡∞, and where the Poisson measure is not compensated. In such a case, the Poisson measure is nite, thus it can be written as N = ∑ i=1 {Ti; Xi ; Zi}, and hence the weak solution of (0.1) is given by V (x; t) = 1 + ∑ i=1 Gt−Ti(x; Xi)1{t¿Ti}: In this case, we see that for each !∈ satisfying (!)¿1, the map t 7→ V (X1(!); t)(!) explodes when t decreases to T1(!). N. Fournier / Stochastic Processes and their Applications 87 (2000) 115–147 119 We are now interested in the Malliavin calculus. We thus suppose some more conditions. First, the intensity measure of N has to be su ciently “regular”. Assumption (M). N has the intensity measure (ds; dy; dz)=’(z)1O(z) ds dy dz, where O is an open subset of R, and ’ is a strictly positive C1 function on O. The functions f, g, h also have to be regular enough. Assumption (H′). f and g are C1 functions on R, and their derivatives are bounded. The function h(x; z) on R × O admits the continuous partial derivatives hz, hx, and h′′ zx = h ′′ xz.There exist a constant K and a function ∈L2(O;’(z) dz) such that for all x∈R, all z ∈O, |hz(0; z)|+ |h′′ xz(x; z)|6K; |h(0; z)|+ |hx(x; z)|6 (z): (1.11) Note that (H′) is stronger than (H). Let be a strictly positive C1 function on O such that and ′ are bounded, and such that ∈L1(O;’(z) dz): (1.12) This “weight function” can be chosen according to the parameters of (0.1). The next condition is technical. Assumption (S). There exists a family of C1 positive functions K on O, with compact support (in O), bounded by 1, and such that ∀z ∈O;K (z)→ →0 1; ∫ O (K ′ (z)) 2 (z) (z)’(z) dz → →0 0: (1.13) We nally suppose one of the following non-degeneracy conditions: Assumption (EW). For all x in R, f(x) 6= 0. Assumption (EP1). f = 0, and there exists ̂∈L1(O;’(z) dz) such that 06hx(x; z)6 ̂(z). For all x in R, ∫ O 1{h′z(x; z)6=0}’(z) dz =∞: (1.14) Assumption (EP2). We set H = {z ∈O=∀x∈R; hz(x; z) 6= 0}. There exist some constants C0¿ 0, r0 ∈ ] 4 ; 1[, and 0¿0 such that for all ¿ 0, ∫ H (1− e− )’(z) dz¿C0 × r0 : (1.15) Our second main result is the next theorem. Theorem 1.5. Assume (M), (D), (H′), and (S). Let V be the unique weak solution of (0:1) in the sense of De nition 1:3; and let (x; t)∈ [0; L] × ]0; T ]. Then under one 120 N. Fournier / Stochastic Processes and their Applications 87 (2000) 115–147 of the assumptions (EW), (EP1) or (EP2), the law of V (x; t) admits a density with respect to the Lebesgue measure on R. We will use two derivative operators. The rst one, associated with the white noise, is classical (see Nualart, 1995). The second operator, associated with the Poisson measure, is inspired from Bichteler et al. (1987, Chapter IV). They study the Malliavin calculus for di usion processes with jumps, in the case where the intensity measure of the Poisson measure is 1O(z) ds dz. Furthermore, they do not use any derivative operator: they work with a “scalar product of derivation”, which gives less information. Using this method, we could probably prove Theorem 1.5 only under (EP1). Our theorem gives in fact two results: the law of V (x; t) admits a density either owing to W or owing to N . It seems to be very di cult to state a “joint” non-degeneracy condition (see Section 3.5). Assumption (EW) looks reasonable: although Pardoux and Zhang prove this theorem under a really less stringent assumption when h= 0 in Pardoux and Zhang (1993) (it su ces that ∃y∈ [0; L] such that f(V0(y)) 6= 0), they use the continuity of their solution. The rst condition in (EP1) (f = 0, hx¿0, h ′ x6̂) is very stringent, but the second one might be optimal: Bichteler et al. also have to assume this kind of condition. Finally, (EP2) is much more general, but it is a uniform non-degeneracy assumption. St Loubert Bi e proves (1998) the existence of a density under the assumption f=0, an hypothesis less stringent than (M), an assumption quite similar to (H′), and under (h1) or (h2) below (the notations are adapted to our context): (h1) hx = 0 and ∫ O 1{h′z(z)=0}’(z) dz =∞ or (h2) ∈L1(O;’(z) dz); hx¿0, and something like (EP1), but depending on the solution process V. Condition (h1) is very restrictive, and (h2) is not very tractable: one has to know the behaviour of the weak solution. Saint Loubert Bi e uses in both case the positivity of N (as in the proof of Theorem 1.5 under (EP1)). However, since the white noise is signed, this method cannot be extended to the case where f 6≡0. That is why in this work, the most interesting assumption is probably (EP2). Let us nally give examples about assumptions (S) and (EP2). Remark 1.6. Assume that O = R. Then (S) is satis ed for any ’; ; and any choice of . Proof. It su ces to choose a family of C1 positive functions of the form K (z) = { 1 if |z|¡ 1= ; 0 if |z|¿ 1= + 2; such that |K ′ (z)|61{|z|∈[1= ;1= +2}. Using the Lebesgue Theorem and the fact that 2 ∈ L1(R; ’(z) dz), (1.13) is immediate. N. Fournier / Stochastic Processes and their Applications 87 (2000) 115–147 121 Example 1. Assume that O = R, and that ’ is a C1 function on R satisfying, for some K ¿a¿ 0, K ¿’¿a. We consider a function h(x; z) = c(x) (z), where c is a strictly positive C1 function on R of which the derivative is bounded. has to be C1 on R, to belong to L2(R; ’(z) dz), and ′ must be bounded. If for some b∈R, [b;∞[⊂{ ′ 6= 0}, then (M), (H′), (S), and (EP2) are satis ed: owing to Remark 1.6, it su ces to check (EP2). Choosing (z)¿z−7=61{z¿b∨1}, we see that (1.15) is satis ed, since [b;∞[⊂H, and using ∀x∈ [0; 1]; 1− e−x¿x=2: (1.16) Example 2. Assume that O=]0; 1[, and that ’(z)=1=zr , for some r ¿ 4 . We consider a function h(x; z)=c(x) (z), where c is a strictly positive C1 function on R the derivative of which is bounded, and where (z)= z , for some ¿ 1∨ (r−1)=2∨ (7− r)=6. Then (M), (H′), (S), and (EP2) are satis ed: (M) is met, and (H′) holds, since ¿ 1∨ (r − 1)=2. It is clearly possible to choose (z) of the form (z) = { z if z6 4 ; (1− z) if z¿ 4 (1.17) with ¿ 1 ∨ (r − 1) and ¿1. Using (1.16), the facts that H = ]0; 1[ and that (z)¿z 1]0;1=4[(z), we see that (EP2) is satis ed if ¡ 3 (r − 1). We now choose a family K of C1 positive functions on ]0; 1[, bounded by 1, and satisfying K (z) =   0 if z¡ =2; 1 if ¡ z¡ 1− ; 0 if 1− =2¡z¡ 1; |K ′ (z)|6 4 1] =2; [∪]1− ;1− =2[(z): An explicit computation shows that (S) is satis ed if ¿r + 1− 2 and if ¿ 1. Since ¿ (7 − r)=6 and r ¿ 4 , it is possible to choose in ](r − 1); 3 (r − 1) c[ ∩ ]1;∞[ ∩ ]r + 1− 2 ;∞[, and the conclusion follows. Let us nally remark that (S) is satis ed for any O, , , if is of class C2 b , and if (z) + | ′(z)| →z→@O 0. Bichteler et al. (1987) assume this kind of condition about . 2. Existence and uniqueness In this short section, we sketch the proof of Theorem 1.4. We begin with a fundamental lemma. Lemma 2.1. Assume (H). Let Y be a process of class PV. Then the processes U (x; t) = ∫ t 0 ∫ L 0 ∫ R Gt−s(x; y)h(Y (y; s); z)Ñ (ds; dy; dz); (2.1)