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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Multiple path-valued
conditional Yeh-Wiener integrals


Authors: Chull Park and David Skoug
Journal: Proc. Amer. Math. Soc. 124 (1996), 2029-2039
MSC (1991): Primary 28C20, 60J65
DOI: https://doi.org/10.1090/S0002-9939-96-03458-2
MathSciNet review: 1342039
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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

\begin{displaymath}F(x)=\exp \left\{\int _0^S\int _0^T\phi (s,t,x(s,t))\,dt\,ds\right \} \end{displaymath}

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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Additional Information

Chull Park
Affiliation: Department of Mathematics & Statistics, Miami University, Oxford, Ohio 45056
Email: cpark@miavxl.acs.muohio.edu

David Skoug
Affiliation: Department of Mathematics & Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: dskoug@unl.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03458-2
Keywords: Yeh-Wiener integral, conditional Yeh-Wiener integral, Wiener integral equation, Cameron-Martin translation theorem
Received by editor(s): December 14, 1994
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society