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 be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of . Then, we use the Fock transform to define some fundamental operators on generalized functionals of , 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 , and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of .
Cite
@article{arxiv.1909.13290,
title = {Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises},
author = {Caishi Wang and Shuai Lin and Ailing Huang},
journal= {arXiv preprint arXiv:1909.13290},
year = {2019}
}