Wick product and stochastic partial differential equations with Poisson measure

We establish the existence and uniqueness of the solution for one conservation equation perturbed by a Poisson noise and when the initial value is afhne. We give also the existence and uniqueness of the solution for a multidimensional linear Skorohod stochastic differential equation driven by a Poisson measure. Resume. En utilisant la theorie de noyaux et symboles sur l’espace de Fock, nous obtenons un resultat d’existence et d’unicité de la solution a une equation de conservation perturbee par un bruit Poissonien et lorsque la condition initiale est affine. Nous donnons aussi un resultat d’existence et d’unicite de la solution a une equation differentielle stochastique lineaire multidimensionnelle au sens de Skorohod et perturbee par un bruit Poissonien.


Introduction
In stochastic analysis on the Wiener space the Wick product is natural because it is implicit in the Ito integral (and, more generally, in the Skorohod integral if the integrand is anticipating).The Skorohod integral coincides with the Ito integral if the integrand is non-anticipating see e.g.Nualart, Pardoux (1988).
Today the Wick product and the Skorohod integral are important in the study of stochastic ordinary and partial differential equations see e.g.Nualart, Zakai   (1989), Holden, Lindstrom, Oksendal, Uboe, Zhang (1994).In the Poisson case some works were also done on this subject Dermoune, Kree, Wu (1988), Carlen,   Pardoux (1988), Nualart, Vives (1990), Dermoune (1995).Here we study one stochastic partial differential equation (SPDE) and one stochastic differential equation (SDE) in order to show that the Wick product also arrives naturally in the Poisson case.The first SPDE is the following conservation equation perturbed by a Poisson measure (see e.g.Whitham (1974)) : ~t( u(x,t)-q(( 0,t])) +~x( au(x,t)2 +bu(x,t)) =0 x ~ IR, t > 0, ( 1) where q(t) = ~s bt; (t) -dt is the centered Poisson measure on 7R+, and a, bare two random variables on Poisson space.8t, as denote the partial derivatives with respect to t and z.Since t -~ q( t) is a distribution and not a smooth function, we interpret the equation (1) in the distribution sense, and the products (au(z, t) are interpreted as the Wick product of generalized random variables.The second SDE concerns the following problem : Let E be a locally compact set, B be the Borel u-field over [0, 1] x E, and v be a positive Radon measure on ([0,1] x E,8) such that v({(t, x)}) = 0, d(t, x) E (0,1~ x E. We denote by q the centered Poisson random measure on [0,1] x E with intensity v, and Ft, 0 t 1, is the natural filtration of q.Consider the SDE of the form where Xo is a d-dimensional random vector, A(s, x) and Bs are d x d deter- ministic matrices.The latter SDE was studied by Buckdahn, Nualart (1994), when the Wiener noise takes the place of the Poisson measure q.If Xo is not a deterministic vector then the process (Xs_ ) is anticipating, and the inte- gral fo fE A(s, x)Xs-dq(s, x) can not be defined in Itô's sense.Nevertheless if A E L~((0,1) x E, v), and B E L1((0,1~, dt) then this integral can be interpreted as a pathwise integral with respect to q, and the equation (2) has a unique pathwise solution Xt = [Id + 01 . . .t n t (~EA(t1, x1)dq(t1,x1) + B(t1)dt1) ...

Xo,
where Id is the identity of d x d matrices.But if A E L~((0,1) x E, v) then the equation (2) has neither pathwise interpretation nor Ito's interpretation.In this case we propose to interpret the stochastic integral fo fE A(s, x)Xs-dq(s, x) in the Skorohod sense given by Dermoune, Kree, Wu (1988), Nualart, Vives (1990), and to study the equation (2) in this sense.Note that if X,-is non-anticipating then the Skorohod integral of (A(s, x)Xs_) coincides with its Itô's integral.But pathwise and the Skorohod interpretations do not coincide.
The plan and the main results of this work are the following : in the section 2 we recall some definitions and results Kree (1986), Dermoune, Kree, Wu (1988), Nualart, Vives (1990), which will be used in the section 3. The combined characteristics and symbol method is used in the first part of the section 3 to establish the existence and the uniqueness of the solution for the SPDE (1) when the initial value is affine.In the second part of section 3 we show that nearly all the results of Buckdahn, Nualart (1994) rest valid for the SDE (2) and we emphasize some differences.
2 Differential calculus relative to the Poisson measure Let X be a locally compact set, B be the Borel u-field over X , and be a positive Radon measure on (X,,~) such that ~(~x}) = 0, dx EX.Let U be a subspace of which is dense in The set S~ :~ Mp(X) is the space of Radon point measures on X.A generic element of U is denoted by z, and a generic element of 03A9 is denoted by w.The duality between Q and U is denoted by j,2r >.We have the triplet U ~ H = L2(X,,~, ~ S~. The scalar product over H is denoted by ., .>, and the norm by ( ~ ~ ~ 2. The triplet (03A9, .P, P) is the probability space of the Poisson measure on (X, B, ).
We denote by L2(~) the space of the square integrable random variables with respect to P, and E denotes the expectation.The centered Poisson measure q is defined by q(z) =

> -fX
It follows from the characteristic functions of P that jE* [|q(z)|2] = ~z~22.From this we see that if z E H, and we choose z" E U such that zn --~ z in H then q(z) := limn-+oo q(zn) in H and the limit is independent of the choice of {zn ~.
Wiener-Itô isomorphism.The symmetric Fock space over H is defined by Fock(H) = ~~k=0Hok, Ho0 = R and for k E IN*, the space Hok is the set of the class of square integrable functions with respect to ~®k, which are symmetric with respect to the k parameters x1, ..., xx.We will denote the norm over also by ~~ ~ The scalar product ., .> over Fock(H) is defined by (fk), (9k) >= fk, gk >.For h E H, we denote by eh the ex- ponential vector, element of Fock(H), defined by eh = ; h~° : = 1.
For k E IN* and for fk E the random variable Ik(fk) is the symmet- ric multiple integral with respect to q defined in Surgailis (1984) and denoted formally by Ik(fk) = fXk fk(x1, ..., xk) dq(xl)...dq(xk).The random variables Fx = k E ~V are such that IE[FkFj ] = 0, for j ~ k. (3) The Wiener-Ito expansion for the centered Poisson measure q means the iso- morphism I from Fock(H) into L~(Q) defined by ( fk) E Fock(H) --~ F = Ik(fk).For z E H n L1(X, p), the image E(z) of the exponential vector e~ by the Wiener-Ito isomorphism I is given by := 7(~)(~) = e-~ J][ (l + (4) j where w = 0 3 A 3 n ( 0 3 C 9 ) j = 1 03B4xj.We can see from ( 3) that for all fix E z E H, _ fk,z~k > . (5) Distributions on (03A9, F, P).Let S(U) be the space of all tensors over U.It is well known that the set S(U)* of linear forms on S(U) is the space of formal series on U, and we have the triplet S(U) -.Fock(H) -S(U)* ( 6) We define the duality between S(U) and S(U)* as an extension of the scalar product over Fock(H), i.e. for all z E U, F = ~~ Fn E S(U)*, F, >_ n!Fn(z).From (6), an element of Fock(H) is interpreted as a formal series on U defined by z E H -~ F, ex >:= F(z).
Generalized random variables.For simplicity we put = The image of S(U) by the isometry I is the space P(O) spanned by k E ~V, and z E U.Moreover, if z E LP(X) then E see proposition 3.1 in Surgailis (1984).The set P(Q)* of linear forms on P(Q) is called the set of generalized random variables.From that we obtain the triplet ~ LZ(~) ~ P(f~)*. (7) The isomorphism 1 from Fock(H) into L2(03A9) is such that I (S(U)) = P(03A9).
Thus the transpose I*, of the restriction of I from S(U) into P(Q), defines an isomorphism from P(S2)* into S(U)*.The Wiener-Ito expansion I is ex- tended to an isomorphism between the triplets ( 6) and ( 7), and we denote it also by l.For A > 0 the space Ha, defined by F = = oo, is a subspace of P(03A9)* .
Gradient operator and divergence operator.In the triplet (6) the gradient operator D is defined from S(U) to S(U)®U by Dzn The transpose DT of D defines the divergence operator from (S(U) @ U)* to S(U)*, and we have for all F E (S(U) ® U)* and z E U the following equality in the formal series sense ~~ a n, zn >_ ~~° o n, F, Dx" >. .
Skorohod integral with respect to the centered Poisson measure.Us- ing the isomorphism I between the triplets ( 6) and ( 7) we define the gradient operator V on and the divergence operator ~ on ® U)*.An element of (P(S~) ® U)* is called a generalized stochastic process.For F E ® U)* we have the generalized integration by parts formula >= F, ~In(zn) >, where ., .> denotes the duality between P(03A9)* and P(03A9).If X = [0,1] x E then the It6 integral with respect to the centered Poisson measure q is extended by the divergence operator 6, Dermoune, Kree, Wu (1988).
Wick product of two generalized random variables.A generalized ran- dom variable F E P(SI)* is characterized by its symbol z E U --> F(z) = F, In(zn) >. .If F, G E P(SI)* then the Wick product FoG of F and G is the element of P(f2)*, defined by Example, i) Let x EX, , the generalized random variable q( x) is defined by the solution of ( 9) is given by t X(y, t, z) = a(z) (t -s)z(s)ds + (a(z)u(y, o, z) + b(z))t + y, (10) 0 and t V(y, t, z) = u(y, o, x) + x(s)ds.
(I1) 0 Thus, the solution u(ac, t, x) of (8) may be described in two different ways: on the one hand we may consider it at a fixed point of space, at time t.This is the so-called Eulerian description.On the other hand, we may follow the wave evolution along the characteristic X(y, t, x), defined by the initial coordinate y.This description is a Lagrangian one, with y the Lagrangian coordinate.To pass from the Lagrangian approach to the Eulerian one it is necessary to find an initial Lagrangian coordinate t) such that x = X(y(z, t), t, x).From (10), ( 11 ) we have u(x, t, x) = u(y, o, x) + z(s)ds, If the initial value is affine then we have the following result.
But if aa 0 then this solution is only defined for 0 t -a'la'1. .
Proof.First, it is well known that a series z --~ S(z), such that S(0) ~ 0, is invertible in S(U)* .From that and from the condition a(o)a(Q) > 0 we conclude that for all t E IR+, z ~ (a(z)a(x)t + 1)'I belongs to S(U)*.It follows that x --~ (a(z)a(x)t + I)'1 is the symbol of a generalized random variable A(a, a, t). .
It is easy to see from (5) that ~t0 sz(s)ds, ~t0 z(s)ds are respectively the symbol of the random variables ~o sdq(s) and q((o, t~).From that and from the definition of the Wick product we have the result.
Example, Suppose that a is deterministic, a ~ q((O,T'~) and T > 0. The symbol A(a, a, t, z) of the generalized random variable A(a, a, t) is given by ~y~ It is easy to prove that ( is the symbol of In(1~n(0,T ]).From that we have A(a, a, t) 0 3 A 3 ñ = 0 ( -t 0 3 B 1 ) n I n ( 1 ñ ( 0 , T ] ) , and does not belong to U03BB>0 Ha.T-At (I1 (f))= I1 ((1+ A1[0,t])-1f)-~t0 ~E A ( s , x ) 1 + ( A ( s , x ) f(s, x)dv(s, x).The operator T~ can be extend to but the estimate of its norm is slightly different from the Wiener case.