Explicit stochastic integral representations for historical functionals. (English) Zbl 0852.60062

Evans, Steven N.; Perkins, Edwin A.

Ann. Probab. 23, No. 4, 1772-1815 (1995).

Summary: 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.

MSC:

60H05	Stochastic integrals
60G57	Random measures
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

superprocess; historical process; martingale measure; stochastic integral; predictable representation; Clark’s formula; Wiener chaos