A simple formula for an analogue of conditional Wiener integrals and its applications

Abstract:

Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ and for a partition $\tau : 0=t_0< t_1< \cdots < t_n=T$ of $[0, T]$, let $X_\tau :C[0,T]\to \mathbb R^{n+1}$ be given by $X_\tau (x) = ( x(t_0), x(t_1), \cdots , x(t_n))$. In this paper, with the conditioning function $X_\tau$, we derive a simple formula for conditional expectations of functions defined on $C[0,T]$ which is a probability space and a generalization of Wiener space. As applications of the formula, we evaluate the conditional expectation of functions of the form \begin{equation*} F_m(x) = \int _0^T (x(t))^m dt, \quad m\in \mathbb N, \end{equation*} for $x\in C[0, T]$ and derive a translation theorem for the conditional expectation of integrable functions defined on the space $C[0,T]$.