Annals of Functional Analysis
- Ann. Funct. Anal.
- Volume 7, Number 2 (2016), 358-370.
Scale transformations for present position-dependent conditional expectations over continuous paths
Let denote a generalized Wiener space, the space of real-valued continuous functions on the interval , and define a random vector by where is a partition of and with almost everywhere. Using a simple formula for a generalized conditional Wiener integral on with the conditioning function , we evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function for , where , is a unit element in , and is the Fourier transform of a measure of bounded variation over . We then express the generalized analytic conditional Feynman integral of as two kinds of limits of nonconditional generalized Wiener integrals with a polygonal function and cylinder functions using a change-of-scale transformation. The choice of a complete orthonormal subset of used in the transformation is independent of .
Ann. Funct. Anal., Volume 7, Number 2 (2016), 358-370.
Received: 25 August 2015
Accepted: 2 November 2015
First available in Project Euclid: 8 April 2016
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 46T12: Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds [See also 28Cxx, 46G12, 60-XX]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
Cho, Dong Hyun. Scale transformations for present position-dependent conditional expectations over continuous paths. Ann. Funct. Anal. 7 (2016), no. 2, 358--370. doi:10.1215/20088752-3544830. https://projecteuclid.org/euclid.afa/1460141561