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

Bong Jin Kim, Byoung Soo Kim


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 $E_\sigma$.


conditional Wiener integral, conditional integral transform, conditional convolution, first variation

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