Asymptotic expansions for functionals of a Poisson random measure

Abstract

The Malliavin calculus for functionals of a Poisson random measure has been developed by many authors. Bismut [2] has generalized the Malliavin calculus for Wiener-Poisson functionals by using the Girsanov theorem. As another method, in Bichteler, Gravereaux and Jacod [1], one can find the study of the Malliavin operator on Wiener-Poisson space and application of it to stochastic differential equations. Both in these works, the authors have given differential operators on Wiener-Poisson space and have proved the integration by parts formulas. These formulation suffers some limitation on an intensity measure, that is, the intensity measure must have a smooth density. On the other hand, in the Malliavin calculus for Wiener functionals, Wiener chaos expansion of the space of square integrable Wiener functionals can be considered as a Fock space, and the differential operator is regarded as the annihilation operator on a Fock space. This sort of structure can be also found in the case of the space of square integrable functionals of Wiener-Poisson space, see [6]. Nualart and Vives [10], [11], and Picard [13] have studied the annihilation operator and its dual operator (the creation operator) on the space of square integrable functionals of a Poisson random measure. Picard [12] has also given a smoothness criterion by using the duality formula (see Theorem 2.1) for functionals of a Poisson random measure under the Condition 1 (see Section 2) on the intensity measure, and has studied the solution to some stochastic differential equation. This argument of Picard can be generalized for some Wiener-Poisson functionals, see [5]. The Condition 1 differs from that of [1], and allows us to take a intensity measure with some singularity. One can find some interesting examples satisfying Condition 1, for instance, stable processes and CGMY processes (see [3]). The purpose of this paper is to prove the asymptotic expansion theorem (done in the Wiener space by Watanabe [18]) for functionals of a Poisson random measure. By using the Malliavin operator which we mentioned above, Sakamoto and Yoshida [15] have studied asymptotic expansion formulas of some