Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

Abstract

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let Z be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of Z. Then, we use the Fock transform to define some fundamental operators on generalized functionals of Z and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of Z and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of Z.