A note on a result of Liptser-Shiryaev

TL;DR

This paper revisits the Liptser-Shiryaev result on the equivalence of laws of solutions to stochastic equations, aiming to weaken the assumptions by focusing solely on the difference of the drift terms, with applications in finite and infinite dimensions.

Contribution

It provides a new version of the Liptser-Shiryaev theorem that requires only the difference of drift terms, simplifying assumptions for law equivalence of stochastic equations.

Findings

Established law equivalence under weaker drift assumptions

Extended results to infinite-dimensional stochastic equations

Provided applications demonstrating practical relevance

Abstract

Given two stochastic equations with different drift terms, under very weak assumptions Liptser and Shiryaev provide the equivalence of the laws of the solutions to these equations by means of Girsanov transform. Their assumptions involve both the drift terms. We are interested in the same result but with the main assumption involving only the difference of the drift terms. Applications of our result will be presented in the finite as well as in the infinite dimensional setting.