Abstract
In this chapter we give the definition of the Poisson measure on a space of configurations of a metric space X, and we construct an isomorphism between the Poisson measure on X and the Poisson process on R+. From this we obtain the probabilistic interpretation of the gradient D as a finite difference operator and the relation between Poisson multiple stochastic integrals and Charlier polynomials. Using the gradient and divergence operators we also derive an integration by parts characterization of Poisson measures, and other results such as deviation and concentration inequalities on the Poisson space.
Keywords
- Poisson Process
- Probabilistic Interpretation
- Stochastic Integral
- Poisson Random Measure
- Poisson Measure
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© 2009 Springer-Verlag Berlin Heidelberg
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Privault, N. (2009). Analysis on the Poisson Space. In: Stochastic Analysis in Discrete and Continuous Settings. Lecture Notes in Mathematics(), vol 1982. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02380-4_7
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DOI: https://doi.org/10.1007/978-3-642-02380-4_7
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02379-8
Online ISBN: 978-3-642-02380-4
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