Poisson imbedding meets the Clark-Ocone formula (2404.07541v1)

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.