Perturbation analysis of Poisson processes

Details Perturbation analysis of Poisson processes Perturbation analysis of Poisson processes

2012-03-14

arxiv.org/abs/1203.3181v1

Mathematics

Probability

Scientific paper

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 ealier results of Molchanov and Zuyev (2000) 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 pure jump 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 (2011).

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