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.