Stochastic analysis for Poisson processes

By Günter Last. 2014

61 citations recorded by the source corpus.

This survey is a preliminary version of a chapter of the forthcoming book [21]. The paper 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 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-Itô 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-Itô 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 Poincaré inequality and the FKG-inequality.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a584e6e7e3a655c6b46857590244708885f9aa28

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Citations in the corpus

Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, background): “1 in [6] implies that the product Dz F Dz L−1 ( F − EF) has a finite chaos expansion with an order less than or equal to 2k − 2.”
Determinantal Point Processes (2016, background): “6) are generalizations of equations (7) and (8) of [21].”
Lectures on the Poisson Process (2017, background): “3 is a slight generalisation of a result in [87].”