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

TL;DR

This paper extends the Clark-Ocone formula to generalized functionals of discrete-time normal noises, broadening its applicability beyond square integrable cases and providing new operator and identity results.

Contribution

It introduces a generalized Clark-Ocone formula for a wider class of functionals of discrete-time normal noises, using the Fock transform and regularity analysis.

Findings

Established the Clark-Ocone formula for generalized functionals of discrete-time normal noises.

Proved the continuity of fundamental operators on these generalized functionals.

Derived covariant identity and upper bound results for the functionals.

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 above mentioned 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 $Z$.