Conditional wiener integral associated with Gaussian processes and applications

Abstract

Let C0[0,T] denote the one-parameter Wiener space and let C?0[0,T] be the Cameron-Martin space in C0[0,T]. Given a function k in C?0[0,T], define a stochastic process Zk : C0[0,T] ? [0, T] ? R by Zk(x, t) = R t 0 Dk(s)dx(s), where Dk ? d/dt k. Let a random vector XG,k : C0[0,T] ? Rn be given by XG,k(x) = ((g1,Zk(x,?))~,..., (gn,Zk(x,?))~), where G = {g1, ..., gn} is an orthonormal set with respect to the weighted inner product induced by the function k on the space C?0[0,T], and (g,Zk(x,?))~ denotes the Paley-Wiener-Zygmund stochastic integral. In this paper, using the reproducing kernel property of the Cameron-Martin space, we establish a very general evaluation formula for expressing conditional generalized Wiener integrals, E(F(Zk(x,?)) |XG,k(x) = ??), associated with the Gaussian processes Zk. As an application, we establish a translation theorem for the conditional Wiener integral and then use it to obtain various conditional Wiener integration formulas on C0[0,T].