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 $Z:C[0,T]\times [0,T]\to \mathbb{R}$ by $Z(x,t)={\int }_0^t\mathrm{‍}h(u)dx(u)+x(0)+a(t)$, for $x\in C[0,T]$ and $t\in [0,T]$, where $h\in L_2[0,T]$ with $h\ne 0$ 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 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 $\text{e}\text{x}\text{p}\{{\int }_0^T\mathrm{‍}Z(x,t)d{m}_L(t)\}$ including the time integral on $C[0,T]$.