Absolute continuity and convergence in variation for distributions of functionals of Poisson point measure

TL;DR

This paper establishes conditions for absolute continuity and convergence in variation of distributions of functionals of Poisson point measures, using time stretching transformations and differential operators, with applications to SDEs driven by Poisson measures.

Contribution

It introduces new sufficient conditions for absolute continuity and convergence in variation for Poisson functionals, utilizing time stretching transformations and differential operators.

Findings

Conditions for absolute continuity are established.

Convergence in variation criteria are provided.

Applications to Poisson-driven SDEs, including non-constant jump rates.

Abstract

General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the unctionals on a probability space, generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form (0,\infty)\times U, and its intensity measure to be equal dt\Pi(du). We introduce the family of time stretching transformations of the configurations of the point measure. The sufficient conditions for absolute continuity and convergence in variation are given in the terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDE's driven by Poisson point measures, including an SDE's with non-constant jump rate.