Korean J. Math. Vol. 24 No. 3 (2016) pp.573-586
DOI: https://doi.org/10.11568/kjm.2016.24.3.573

Conditional integral transforms and convolutions for a general vector-valued conditioning functions

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Bong Jin Kim
Byoung Soo Kim

Abstract

We study the conditional integral transforms and conditional convolutions of functionals defined on $K[0, T]$. We consider a general vector-valued conditioning functions $X_k (x) = \left(\gamma_1 (x), \ldots, \gamma_k (x)\right)$ where $\gamma_j (x)$ are Gaussian random variables on the Wiener space which need not depend upon the values of $x$ at only finitely many points in $(0, T]$. We then obtain several relationships and formulas for the conditioning functions that exist among conditional integral transform, conditional convolution and first variation of functionals in $E_\sigma$.


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