Malliavin calculus for parabolic SPDEs with jumps

Abstract

We study a parabolic SPDE driven by a white noise and a compensated Poisson measure. We first define 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.