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

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 \(Z\).