Multiple path-valued conditional Yeh-Wiener integrals

Park, Chull; Skoug, David

doi:10.1090/S0002-9939-96-03458-2

Multiple path-valued conditional Yeh-Wiener integrals
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Proc. Amer. Math. Soc. 124 (1996), 2029-2039 Request permission

Abstract:

In this paper we establish various results involving parallel line-valued conditional Yeh-Wiener integrals of the type $E(F(x)|x(s_j,\boldsymbol {\cdot })=\eta _j(\boldsymbol {\cdot })$, $j=1,\dotsc ,n)$ where $0<s_1<\cdots <s_n$. We then develop a formula for converting these multiple path-valued conditional Yeh-Wiener integrals into ordinary Yeh-Wiener integrals. Next, conditional Yeh-Wiener integrals for functionals $F$ of the form \[ F(x)=\exp \left \{\int _0^S\int _0^T\phi (s,t,x(s,t)) dt ds\right \} \] are evaluated by solving an appropriate Wiener integral equation. Finally, a Cameron-Martin translation theorem is obtained for these multiple path-valued conditional Yeh-Wiener integrals.

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