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

Abstract:

In this paper we evaluate the conditional Yeh-Wiener integral $E(F(x)|x(s,t) = \xi )$ for functions $F$ of the form \[ F(x) = \exp \{ \int _0^t {\int _0^s \phi } (\sigma ,\tau ,x(\sigma ,\tau ))d\sigma d\tau \} .\] The method we use to evaluate this conditional integral is to first define a sample path-valued conditional Yeh-Wiener integral of the type $E(F(x)|x(s,) = \psi ())$ and show that it satisfies a Wiener integral equation. We next obtain a series solution for $E(F(x)|x(s,) = \psi ())$ by solving this Wiener integral equation. Finally, we integrate this series solution appropriately in order to evaluate $E(F(x)|x(s,t) = \xi )$.