An operator-valued Yeh-Wiener integral and a Kac-Feynman Wiener integral equation

Park, Chull; Skoug, David

doi:10.1090/S0002-9939-1994-1213867-1

An operator-valued Yeh-Wiener integral and a Kac-Feynman Wiener integral equation
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by Chull Park and David Skoug PDF

Proc. Amer. Math. Soc. 120 (1994), 929-942 Request permission

Abstract:

Let $C[0,T]$ denote Wiener space, i.e., the space of all continuous functions $\eta (t)$ on $[0,T]$ such that $\eta (0) = 0$. For $Q = [0,S] \times [0,T]$ let $C(Q)$ denote Yeh-Wiener space, i.e., the space of all $\mathbb {R}$-valued continuous functions $x(s,t)$ on $Q$ such that $x(0,t) = x(s,0) = 0$ for all $(s,t)$ in $Q$. For $h \in {L_2}(Q)$ let $Z(x;s,t)$ be the Gaussian process defined by the stochastic integral \[ Z(x;s,t) = \int _0^t {\int _0^s {h(u,v)dx(u,v).} } \] Then for appropriate functionals $F$ and $\psi$ we show that the operator-valued function space integral \[ (I_\lambda ^h(F)\psi )(\eta ( \cdot )) = {E_x}[F({\lambda ^{ - 1/2}}Z(x; \cdot , \cdot ) + \eta ( \cdot ))\psi ({\lambda ^{ - 1/2}}Z(x;S, \cdot ) + \eta ( \cdot ))]\] is the unique solution of a Kac-Feynman Wiener integral equation. We also use this integral equation to evaluate various Yeh-Wiener integrals.

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Stochastic processes and the Wiener integral