Pathwise uniqueness and continuous dependence for SDEs with nonregular drift

TL;DR

This paper presents a new proof establishing pathwise uniqueness and continuous dependence on initial conditions for stochastic differential equations with irregular drift, using heat equation regularity theory.

Contribution

It introduces a novel proof technique for SDEs with nonregular drift, relying solely on heat equation regularity, enhancing understanding of solution uniqueness.

Findings

Pathwise uniqueness holds for SDEs with certain integrability drift.

Continuous dependence on initial conditions is established despite poor drift regularity.

The proof leverages heat equation regularity theory as the main tool.

Abstract

A new proof of a pathwise uniqueness result of Krylov and R\"{o}ckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.