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

Authors:	Dong Hyun Cho

Institution:	Department of Mathematics, Kyonggi University, Kyonggido Suwon 443-760, Korea

Abstract:	Let denote the space of real-valued continuous functions on the interval and for a partition $\tau: 0=t_0< t_1< \cdots < t_n=T$ of , let $X_\tau:C0,T]\to \mathbb{R}^{n+1}$ 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 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 $\displaystyle F_m(x) = \int_0^T (x(t))^m dt, \quad m\in\mathbb{N},$ for $x\in C0, T]$ and derive a translation theorem for the conditional expectation of integrable functions defined on the space .

Keywords:	Analogue of Wiener measure conditional Cameron-Martin translation theorem conditional Wiener integral simple formula for conditional $w_\varphi$-integral

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