Pathwise uniqueness of non-uniformly elliptic SDEs with rough coefficients

TL;DR

This paper reviews and enhances pathwise uniqueness results for certain one-dimensional SDEs with rough, possibly vanishing diffusion coefficients, using local time comparison and occupation time formulas, leading to stronger solution properties.

Contribution

It introduces new techniques based on local time comparison to establish pathwise uniqueness for SDEs with non-smooth, vanishing diffusion coefficients.

Findings

Pathwise uniqueness holds under broader conditions.

Strong solutions exist for SDEs with rough coefficients.

Methods extend to SDEs with degenerate diffusion coefficients.

Abstract

In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.