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Journal of Probability and Statistics
Volume 2017, Article ID 8510782, 13 pages
https://doi.org/10.1155/2017/8510782
Research Article

A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Department of Mathematics, Kyonggi University, Suwon 16227, Republic of Korea

Correspondence should be addressed to Dong Hyun Cho; rk.ca.iggnoyk@58349j

Received 29 July 2016; Accepted 24 November 2016; Published 28 February 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Dong Hyun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of . We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.