Absolute Continuity and Convergence in Variation for Distributions of Functionals of Poisson Point Measure

Abstract

General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the functionals on the probability space generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form ${\mathbb{R}}^{+}\times{\mathbb{U}}$ , and its intensity measure to equal dt Π(du). We introduce the family of time stretching transformations of the configurations of the point measure. Sufficient conditions for absolute continuity and convergence in variation are given in terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDEs driven by Poisson point measures, including SDEs with non-constant jump rate.