Annals of Functional Analysis

Scale transformations for present position-dependent conditional expectations over continuous paths

Dong Hyun Cho

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Abstract

Let C[0,t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,t], and define a random vector Zn:C[0,t]Rn by Zn(x)=(0t1h(s)dx(s),,0tnh(s)dx(s)), where 0<t1<<tn=t is a partition of [0,t] and hL2[0,t] with h0 almost everywhere. Using a simple formula for a generalized conditional Wiener integral on C[0,t] with the conditioning function Zn, we evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function G(x)=f((e,x))ϕ((e,x)) for xC[0,t], where fLp(R)(1p), e is a unit element in L2[0,t], and ϕ is the Fourier transform of a measure of bounded variation over R. We then express the generalized analytic conditional Feynman integral of G 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 L2[0,t] used in the transformation is independent of e.

Article information

Source
Ann. Funct. Anal., Volume 7, Number 2 (2016), 358-370.

Dates
Received: 25 August 2015
Accepted: 2 November 2015
First available in Project Euclid: 8 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1460141561

Digital Object Identifier
doi:10.1215/20088752-3544830

Mathematical Reviews number (MathSciNet)
MR3484389

Zentralblatt MATH identifier
1346.46038

Subjects
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]

Keywords
analytic conditional Feynman integral analytic conditional Wiener integral conditional Wiener integral Wiener integral Wiener space

Citation

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


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