Grid-valued conditional Yeh-Wiener integrals and a Kac-Feynman Wiener integral equation. (English) Zbl 0863.28007
Summary: We establish several results involving grid-valued conditional Yeh-Wiener integrals of the type
\[
E(F(x)|x(s_1,\cdot),\dots, x(s_m,\cdot), x(*,t_1),\dots,x(*,t_n)).
\]
We develop a formula for converting these grid-valued conditional Yeh-Wiener integrals into ordinary Yeh-Wiener integrals. We also obtain a Cameron-Martin translation theorem for these integrals. More importantly, we evaluate these conditional expectations for functionals \(F\) of the form
\[
F(x)=\exp\Biggl\{\int^T_0\int^S_0\phi(u,v,x(u,v))du dv\Biggr\}
\]
by solving a Kac-Feynman type Wiener integral equation.
MSC:
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
| 60J65 | Brownian motion |
Keywords:
Gaussian process; grid-valued conditional Yeh-Wiener integrals; Cameron-Martin translation theorem; Kac-Feynman type Wiener integral equationReferences:
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