Analogues of Conditional Wiener Integrals with Drift and Initial Distribution 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 Z:C[0,T]×[0,T]→R by Z(x,t)=∫0t‍h(u)dx(u)+x(0)+a(t), for x∈C[0,T] and t∈[0,T], where h∈L2[0,T] with h≠0 a.e. and a is a continuous function on [0,T]. Let Zn:C[0,T]→Rn+1 and Zn+1:C[0,T]→Rn+2 be given by Zn(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn)) and Zn+1(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn),Z(x,tn+1)), where 0=t0<t1<⋯<tn<tn+1=T is a partition of [0,T]. In this paper we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,T] with the conditioning functions Zn and Zn+1 which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function exp{∫0T‍Z(x,t)dmL(t)} including the time integral on C[0,T].