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

Abstract EN

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.