Vol. 59, No. 2, 1975

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Inversion of conditional Wiener integrals

James Juei-Chin Yeh

Vol. 59 (1975), No. 2, 623–638

Given two Wiener measurable functionals X and Y on the Wiener space C[0,t], of which the latter is Wiener integrable, the conditional Wiener integral of Y given X is defined as the conditional expectation Ew(Y |X) given as a function on the value space of X. Several Fourier inversion formulae for retrieving the conditional Wiener integral Ew(Y |X) in which X[x] = x(t) for x C[0,t] are derived. Examples of evaluation of Ew(Y |X) are given. It is shown that the Kac-Feynman formula can be derived by applying an inversion formula to Ew(Y |X) where

Y[x] = exp{− 0 V [x(s)]ds}.

Mathematical Subject Classification
Primary: 28A40
Received: 12 May 1975
Published: 1 August 1975
James Juei-Chin Yeh