Girsanov identities for Poisson measures under quasi-nilpotent transformations

TL;DR

This paper establishes Girsanov identities for Poisson measures under certain anticipating transformations, enabling new insights into measure invariance and transformations in stochastic Poisson spaces.

Contribution

It introduces a Girsanov identity for Poisson measures with quasi-nilpotent transformations, expanding the theoretical framework for stochastic analysis on Poisson spaces.

Findings

Proves a Girsanov identity for Poisson measures under quasi-nilpotent transformations.

Demonstrates invariance of Poisson measures under specific random transformations.

Provides combinatorial identities for moments of Poisson stochastic integrals.

Abstract

We prove a Girsanov identity on the Poisson space for anticipating transformations that satisfy a strong quasi-nilpotence condition. Applications are given to the Girsanov theorem and to the invariance of Poisson measures under random transformations. The proofs use combinatorial identities for the central moments of Poisson stochastic integrals.