Stochastic analysis for Poisson processes

TL;DR

This survey introduces foundational stochastic analysis tools for Poisson processes, including chaos expansions, Malliavin calculus, and covariance inequalities, providing a comprehensive theoretical framework for future research.

Contribution

It develops the basic theory of stochastic analysis for Poisson processes, including chaos expansions, Wiener-It extsuperscript{o} integrals, and Malliavin operators, in a general measure space setting.

Findings

Chaos expansion of square-integrable Poisson functionals

Introduction of multivariate Wiener-It extsuperscript{o} integrals and properties

Derivation of covariance identities and inequalities

Abstract

This survey is a preliminary version of a chapter of the forthcoming book "Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-It\^o Chaos Expansions and Stochastic Geometry" edited by Giovanni Peccati and Matthias Reitzner. The paper develops some basic theory for the stochastic analysis of Poisson process on a general $\sigma$-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate stochastic Wiener-It\^o integrals and the discussion of their basic properties. The paper then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener-It\^o integrals and Mehler's formula for the inverse of the Ornstein-Uhlenbeck generator based on a dynamic thinning procedure. The survey concludes with covariance identities, the Poincar\'e inequality and the FKG-inequality.