# Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes

## With Emphasis on the Creation-Annihilation Techniques

Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 76)

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Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 76)

A simplified approach to Malliavin calculus adapted to Poisson random measures is developed and applied in this book. Called the “lent particle method” it is based on perturbation of the position of particles. Poisson random measures describe phenomena involving random jumps (for instance in mathematical finance) or the random distribution of particles (as in statistical physics). Thanks to the theory of Dirichlet forms, the authors develop a mathematical tool for a quite general class of random Poisson measures and significantly simplify computations of Malliavin matrices of Poisson functionals. The method gives rise to a new explicit calculus that they illustrate on various examples: it consists in adding a particle and then removing it after computing the gradient. Using this method, one can establish absolute continuity of Poisson functionals such as Lévy areas, solutions of SDEs driven by Poisson measure and, by iteration, obtain regularity of laws. The authors also give applications to error calculus theory. This book will be of interest to researchers and graduate students in the fields of stochastic analysis and finance, and in the domain of statistical physics. Professors preparing courses on these topics will also find it useful. The prerequisite is a knowledge of probability theory.

60H07, 60G57, 60G51, 60J45 Dirichlet forms Lévy processes Malliavin calculus lent particle method random Poisson measures

- DOI https://doi.org/10.1007/978-3-319-25820-1
- Copyright Information Springer International Publishing Switzerland 2015
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-25818-8
- Online ISBN 978-3-319-25820-1
- Series Print ISSN 2199-3130
- Series Online ISSN 2199-3149
- Buy this book on publisher's site