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Sample path-valued conditional Yeh-Wiener integrals and a Wiener integral equation

Authors: Chull Park and David Skoug
Journal: Proc. Amer. Math. Soc. 115 (1992), 479-487
MSC: Primary 28C20; Secondary 60J65
MathSciNet review: 1104401
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Abstract: In this paper we evaluate the conditional Yeh-Wiener integral $ E(F(x)\vert x(s,t) = \xi )$ for functions $ F$ of the form

$\displaystyle F(x) = \exp \{ \int_0^t {\int_0^s \phi } (\sigma ,\tau ,x(\sigma ,\tau ))d\sigma d\tau \} .$

The method we use to evaluate this conditional integral is to first define a sample path-valued conditional Yeh-Wiener integral of the type $ E(F(x)\vert x(s,) = \psi ())$ and show that it satisfies a Wiener integral equation. We next obtain a series solution for $ E(F(x)\vert x(s,) = \psi ())$ by solving this Wiener integral equation. Finally, we integrate this series solution appropriately in order to evaluate $ E(F(x)\vert x(s,t) = \xi )$.

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Keywords: Yeh-Wiener integral, conditional Yeh-Wiener integral, Wiener integral equation
Article copyright: © Copyright 1992 American Mathematical Society

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