Wiener Functionals as Itô Integrals

Abstract

Let W(t, ω) be a standard Wiener process, W _t ≡ W(t) ≡ W(t, ∙). A function φ(t, ω) is called nonanticipating iff for all t ^ 0, φ(t, ∙) is measurable with respect to {W _s : 0 % s % t}. The Itô stochastic integral

$$f(\omega )\, \equiv \,\int {_0^1 \varphi (t,\,\omega )d_t \,W(t,\,\omega )}$$

is defined for any jointly measurable, nonanticipating φ such that for almost all $\Omega, \int_0^1 \Phi^2(t, \Omega)dt < \infty$ (Gikhman and skorokhod (1968), Chapter 1, Section 2). It is known that it is defined for any jointly measurable, nonanticipating φ such that for almost all $E \int_0^1 \Phi^2(t, \Phi) dt < \infty, {\rm then} Ef = 0 {\rm and} Ef^2 < \infty.$. Representation of an arbitrary measurable f as a stochastic integral was stated, but later retracted, by J.M.C. Clark (1970, 1971).

Received January 19, 1976.

This research was supported in part by National Science Foundation Grant MPS71-02972 A04.

AMS 1970 subject classifications. Primary 60H05; Secondary 60G15, 60G17, 60G40, 60J65.