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Preprints, Working Papers, ... Year : 2024

Poisson imbedding meets the Clark-Ocone formula *

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 repre- sentation 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 pre- dictable integrable process.
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Dates and versions

hal-04554378 , version 1 (22-04-2024)

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Caroline Hillairet, Thomas Peyrat, Anthony Réveillac. Poisson imbedding meets the Clark-Ocone formula *. 2024. ⟨hal-04554378⟩
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