Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 3, Pages 239-246
doi:10.1155/S1048953394000249

On transformations of Wiener space

Anatoli V. Skorokhod1,2

1Ukranian Academy of Science, Institute of Mathematics, Kiev, Ukraine
2Michigan State University, Department of Mathematics, East Lansing 48824, MI, USA

Received 1 February 1994; Revised 1 April 1994

Copyright © 1994 Anatoli V. Skorokhod. 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

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]→ℝ, x0=0, where a(s,x) is a measurable function [0,1]×C→ℝ which is 𝒞˜s-measurable for a fixed s and 𝒞˜s is the σ-algebra generated by {xu,u≤t}. 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.