Persistence of invertibility on the Wiener space

TL;DR

This paper investigates conditions for the almost sure invertibility of adapted perturbations of the identity on Wiener space, characterizes measure-preserving maps, and applies results to stochastic differential equations involving stopping times.

Contribution

It provides necessary and sufficient conditions for invertibility of adapted perturbations on Wiener space and characterizes measure-preserving maps via innovation processes.

Findings

Characterization of invertibility conditions

Measure-preserving maps identified by innovation processes

Invertibility preserved under stopping time modifications

Abstract

Let $(W,H,\mu)$ be the classical Wiener space, assume that $U=I_W+u$ is an adapted perturbation of identity where the perturbation $u$ is an equivalence class w.r.to the Wiener measure. We study several necessary and sufficient conditions for the almost sure invertibility of such maps. In particular the subclass of these maps who preserve the Wiener measure are characterized in terms of the corresponding innovation processes. We give the following application: let $U$ be invertible and let $\tau$ be stopping time. Define $U^\tau$ as $I_W+u^\tau$ where $u^\tau$ is given by $$ u^\tau(t,w)=\int_0^t 1_{[0,\tau(w)]}(s)\dot{u}_s(w)ds\,. $$ We prove that $U^\tau$ is also almost surely invertible. Note that this has immediate applications to stochastic differential equations.