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

Abstract

LetC[0,t]denote a generalized Wiener space, the space of real-valued continuous functions on the interval[0,t]and define a stochastic processY:C[0,t]×[0,t]→ℝbyY(x,s)=∫0s‍h(u)dx(u)+a(s)forx∈C[0,t]ands∈[0,t], whereh∈L2[0,t]withh≠0a.e. andais continuous on[0,t]. Let random vectorsYn:C[0,t]→ℝnandYn+1:C[0,t]→ℝn+1be given byYn(x)=(Y(x,t1),…,Y(x,tn))andYn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)), where0<t1<⋯<tn<tn+1=tis a partition of[0,t]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove thatYis a generalized Brownian motion process with drifta. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions onC[0,t]with the drift and the conditioning functionsYnandYn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions onC[0,t].