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A simple formula for an analogue of conditional Wiener integrals and its applications
Authors:Dong Hyun Cho
Affiliation:Department of Mathematics, Kyonggi University, Kyonggido Suwon 443-760, Korea
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

$displaystyle F_m(x) = int_0^T (x(t))^m dt, quad minmathbb{N}, $

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

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