Analogues of Conditional Wiener Integrals with Drift and Initial Distribution 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 processZ:C[0,T]×[0,T]→RbyZ(x,t)=∫0t‍h(u)dx(u)+x(0)+a(t), forx∈C[0,T]andt∈[0,T], whereh∈L2[0,T]withh≠0a.e. andais a continuous function on[0,T]. LetZn:C[0,T]→Rn+1andZn+1:C[0,T]→Rn+2be given byZn(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn))andZn+1(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn),Z(x,tn+1)), where0=t0<t1<⋯<tn<tn+1=Tis a partition of[0,T]. In this paper we derive two simple formulas for generalized conditional Wiener integrals of functions onC[0,T]with the conditioning functionsZnandZn+1which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the functionexp{∫0T‍Z(x,t)dmL(t)}including the time integral onC[0,T].