The Clark–Ocone formula for vector valued Wiener functionals

Abstract

The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron–Martin space. The Itô integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense. A complete characterization of measure preserving transformations in Wiener space is presented as an application of this generalized Clark–Ocone formula.