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

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

$$F_m(x) = \int_0^T (x(t))^m dt, \quad m\in\mathbb{N},$$

for $x\in C[0, T]$ and derive a translation theorem for the conditional expectation of integrable functions defined on the space $C[0,T]$.