International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1994 / Article

Open Access

Volume 7 |Article ID 584613 | https://doi.org/10.1155/S1048953394000249

Anatoli V. Skorokhod, "On transformations of Wiener space", International Journal of Stochastic Analysis, vol. 7, Article ID 584613, 8 pages, 1994. https://doi.org/10.1155/S1048953394000249

On transformations of Wiener space

Received01 Feb 1994
Revised01 Apr 1994

Abstract

We consider transformations of the form (Tax)t=xt+0ta(s,x)ds on the space C of all continuous functions x=xt:[0,1], x0=0, where a(s,x) is a measurable function [0,1]×C which is 𝒞˜s-measurable for a fixed s and 𝒞˜s is the σ-algebra generated by {xu,ut}. It is supposed that Ta maps the Wiener measure μ0 on (C,𝒞˜s) into a measure μa which is equivalent with respect to μ0. We study some conditions of invertibility of such transformations. We also consider stochastic differential equations of the form dy(t)=dw(t)+a(t,y(t))dt, y(0)=0 where w(t) is a Wiener process. We prove that this equation has a unique strong solution if and only if it has a unique weak solution.

Copyright © 1994 Hindawi Publishing Corporation. 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.


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