Stochastic Analysis for Poisson Point Processes pp 1-36

Part of the Bocconi & Springer Series book series (BS, volume 7)

Stochastic Analysis for Poisson Processes

Chapter

Abstract

This chapter develops some basic theory for the stochastic analysis of Poisson process on a general σ-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the chapter 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–Itô integrals and the discussion of their basic properties. The chapter then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener–Itô integrals and Mehler’s formula for the inverse of the Ornstein–Uhlenbeck generator based on a dynamic thinning procedure. The chapter concludes with covariance identities, the Poincaré inequality, and the FKG-inequality.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyInstitute of StochasticsKarlsruheGermany

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