Inversion of conditional Wiener integrals

Abstract

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 E w (Y | X) given as a function on the value space of X. Several Fourier inversion formulae for retrieving the conditional Wiener integral E W (Y\X) in which X[x] = x(t) for xeC[0, t] are derived. Examples of evaluation of E W (Y\X) are given. It is shown that the Kac-Feynman formula can be derived by applying an inversion formula to E W {Y\X) where ΓM-expi-0 Consider the Wiener integrable functional exp ί-Γ F[φ)]ώs} for x e C[0, t]

A simple formula for an analogue of conditional Wiener integrals and its applications

“…On the space, Yeh introduced an inversion formula that a conditional expectation can be found by Fourier-transform ( [9]). Also, in [10,11], he obtained very useful results including the Kac-Feynman integral equation and the conditional Cameron-Martin translation theorem using the inversion formula. But Yeh's inversion formula is very complicated in its applications when the conditioning function is vector-valued.…”

Parts Formulas Involving Conditional Integral Transforms on Function Space

“…Then for η ∈ R, E x [F X](η) denotes the conditional Wiener integral of F given X [6,12,16]. In [12], Park and Skoug gave a formula for expressing conditional Wiener integrals in terms of ordinary(i.e., non-conditional) Wiener integrals; namely that for X(x) = x(T ),…”

Conditional Analytic Feynman Integrals on Abstract Wiener Spaces

“…The function if/(cf), <f e R" is unique up to Borel null sets in R" . Following Yeh [13] the function y/(£,), written E(Y\X)(cl), is called the conditional abstract Wiener integral of Y given X (see [4]). …”