Conditional integral transforms of functionals on a function space of two variables

Abstract

Let $C(Q)$ denote Yeh-Wiener space, the space of all real-valued continuous functions $x(s,t)$ on $Q\equiv [0,S]\times [0,T]$ with $x(s,0)=x(0,t)=0$ for every $(s,t)\in Q$. For each partition $\tau ={\tau }_{m,n}=\{(s_i,t_j)|i=1,\dots ,m,j=1,\dots ,n\}$ of $Q$ with $0=s_0<s_1<\dots <s_m=S$ and $0=t_0<t_1<\dots <t_n=T$, define a random vector $X_{\tau }:C(Q)\to {\mathbb{R}}^{mn}$ by $$X_{\tau }(x)=(x(s_1,t_1),\dots ,x(s_m,t_n)).$$

In this paper we study the conditional integral transform and the conditional convolution product for a class of cylinder type functionals defined on $K(Q)$ with a given conditioning function $X_{\tau }$ above, where $K(Q)$is the space of all complex valued continuous functions of two variables on $Q$ which satify $x(s,0)=x(0,t)=0$ for every $(s,t)\in Q$. In particular we derive a useful equation which allows to calculate the conditional integral transform of the conditional convolution product without ever actually calculating convolution product or conditional convolution product.