Explicit Stochastic Integral Representations for Historical Functionals

Abstract

It is known from previous work of the authors that any square-integrable functional of a superprocess may be represented as a constant plus a stochastic integral against the associated orthogonal martingale measure. Here we give, for a large class of such functionals, an explicit description of the integrand that is analogous to Clark's formula for the representation of certain Brownian functionals. As a consequence, we develop a partial analogue of the Wiener chaos expansion in the superprocess setting. Rather than work with superprocesses per se, our results are stated and proved in the richer and more natural context of the associated historical process.