Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space

Abstract

Let C[0,t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,t] and define a stochastic process Y:C[0,t]×[0,t]→ℝ by Y(x,s)=∫0s‍h(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t], where h∈L2[0,t] with h≠0 a.e. and a is continuous on [0,t]. Let random vectors Yn:C[0,t]→ℝn and Yn+1:C[0,t]→ℝn+1 be given by Yn(x)=(Y(x,t1),…,Y(x,tn)) and Yn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)), where 0<t1<⋯<tn<tn+1=t is a partition of [0,t]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove that Y is a generalized Brownian motion process with drift a. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,t] with the drift and the conditioning functions Yn and Yn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on C[0,t].