Perturbation analysis of Poisson processes

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 Lévy processes under perturbations of the Lévy 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].