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arXiv · 2404.07541

Poisson imbedding meets the Clark-Ocone formula

Caroline Hillairet·Thomas Peyrat·Anthony Réveillac
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Abstract

In this paper we develop a representation formula of Clark-Ocone type for any integrable Poisson functionals, which extends the Poisson imbedding for point processes. This representation formula differs from the classical Clark-Ocone formula on three accounts. First the representation holds with respect to the Poisson measure instead of the compensated one; second the representation holds true in L1 and not in L2; and finally contrary to the classical Clark-Ocone formula the integrand is defined as a pathwise operator and not as a L2-limiting object. We make use of Malliavin's calculus and of the pseudo-chaotic decomposition with uncompensated iteraded integrals to establish this Pseudo-Clark-Ocone representation formula and to characterize the integrand, which turns out to be a predictable integrable process.

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Caroline Hillairet, Thomas Peyrat, Anthony Réveillac. 2024-04-11. Poisson imbedding meets the Clark-Ocone formula. https://arxiv.org/abs/2404.07541

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