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Transactions of the American Mathematical Society

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

Author: Dong Hyun Cho
Journal: Trans. Amer. Math. Soc. 360 (2008), 3795-3811
MSC (2000): Primary 28C20
Published electronically: January 30, 2008
MathSciNet review: 2386246
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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 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]$.

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Additional Information

Dong Hyun Cho
Affiliation: Department of Mathematics, Kyonggi University, Kyonggido Suwon 443-760, Korea

Keywords: Analogue of Wiener measure, conditional Cameron-Martin translation theorem, conditional Wiener integral, simple formula for conditional $w_\varphi$-integral
Received by editor(s): May 30, 2006
Published electronically: January 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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