A non-compensated Clark–Ocone formula for functionals of counting processes

¹ ENSAE Paris, CREST UMR 9194, 5 avenue Henry Le Chatelier 91120 Palaiseau, France
² ENSAE Paris, CREST UMR 9194, and Exiom Parners, 26 rue Notre Dame des Victoires, 75002 Paris, France
³ INSA de Toulouse, IMT UMR CNRS 5219, Université de Toulouse, 135 avenue de Rangueil 31077 Toulouse Cedex 4 France

^* Corresponding author: caroline.hillairet@ensae.fr

Received: 10 April 2024
Accepted: 4 February 2025

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 a decomposition with uncompensated iterated integrals derived in [Hillairet and Réveillac, Electron. J. Probab. 29 (2024) 1–33] to establish this non-compensated Clark–Ocone representation formula and to characterize the integrand, which turns out to be a predictable integrable process.