Analogues of conditional Wiener integrals with drift and initial distribution on a function space (Q1725264)

Summary: 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 \(Z: C[0,T]\times[0,T]\to\mathbb R\) by \(Z(x,t)=\int^t_0 h(u)dx(u)+x(0)+a(t)\), for \(x\in C[0,T]\) and \(t\in[0,T]\), where \(h\in L_2[0,T]\) with \(h\neq 0\) a.e. and \(a\) is a continuous function on \([0, T]\). Let \(Z_n: C[0,T]\to\mathbb R^{n+1}\) and \(Z_{n+1}: C[0,T]\to\mathbb R^{n+2}\) be given by \(Z_n(x)=(Z(x,t_0),Z(x,t_1),\dots,Z(x,t_n))\) and \(Z_{n+1}(x)=(Z(x,t_0),Z(x,t_1),\dots,Z(x,t_n),Z(x,t_{n+1}))\), where \(0=t_0<t_1<\dots <t_n<t_{n+1}=T\) 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 \(Z_n\) and \(Z_{n+1}\) which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function \(\exp\{\int^T_0Z(x,t)dm_L(t)\}\) including the time integral on \(C[0,T]\).