Stochastic analysis for Poisson processes

@inproceedings{Last2014StochasticAF,
  title={Stochastic analysis for Poisson processes},
  author={Gunter Last},
  year={2014}
}
  • G. Last
  • Published 17 May 2014
  • Mathematics
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… 
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29 References

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with…

Orthogonal functionals of the Poisson process

  • H. Ogura
  • Mathematics
    IEEE Trans. Inf. Theory
  • 1972
It is shown that any nonlinear functional of the Poisson process with finite variance can be developed in terms of these orthogonal functionals, corresponding to the Cameron-Martin theorem in the case of the Brownian-motion process.

Gamma limits and U-statistics on the Poisson space

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  • N. CutlandSiu-Ah Ng
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1991
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