On Uniqueness for some non-Lipschitz SDE

TL;DR

This paper investigates the uniqueness of solutions to certain stochastic differential equations with additive noise and non-Lipschitz drift, using a path-by-path approach that combines ODE theory and probabilistic methods.

Contribution

It establishes uniqueness results for non-Lipschitz SDEs in a path-by-path sense, extending classical theory to broader classes of noise and drift functions.

Findings

Uniqueness holds for Brownian motion noise.

Applicable to unbounded and discontinuous drift functions.

Uses Girsanov's theorem to prove results.

Abstract

We study the uniqueness in the path-by-path sense (i.e. $\omega$-by-$\omega$) of solutions to stochastic differential equations with additive noise and non-Lipschitz autonomous drift. The notion of path-by-path solution involves considering a collection of ordinary differential equations and is, in principle, weaker than that of a strong solution, since no adaptability condition is required. We use results and ideas from the classical theory of ode's, together with probabilistic tools like Girsanov's theorem, to establish the uniqueness property for some classes of noises, including Brownian motion, and some drift functions not necessarily bounded nor continuous.