Perturbation analysis of Poisson processes

TL;DR

This paper analyzes how expectations of functions of Poisson processes change under perturbations of their intensity measure, extending previous results and applying the findings to Lévy processes.

Contribution

It extends Margulis-Russo type formulas to general Poisson processes with signed perturbations of the intensity measure.

Findings

Derived general formulas for derivatives of Poisson functionals under measure perturbations

Applied results to Lévy processes with perturbed Lévy measures

Provided explicit Fock space representations for analysis

Abstract

We consider a Poisson process $\Phi$ on a general phase space. The expectation of a function of $\Phi$ can be considered as a functional of the intensity measure $\lambda$ of $\Phi$. Extending earlier results of Molchanov and Zuyev [Math. Oper. Res. 25 (2010) 485-508] on finite Poisson processes, we study the behaviour of this functional under signed (possibly infinite) perturbations of $\lambda$. In particular, we obtain general Margulis-Russo type formulas for the derivative with respect to non-linear transformations of the intensity measure depending on some parameter. As an application, we study the behaviour of expectations of functions of multivariate L\'evy processes under perturbations of the L\'evy measure. A key ingredient of our approach is the explicit Fock space representation obtained in Last and Penrose [Probab. Theory Related Fields 150 (2011) 663-690].