Discontinuous Stochastic Differential Equations Driven by L\'evy Processes

TL;DR

This paper establishes pathwise uniqueness for a class of stochastic differential equations driven by symmetric alpha-stable processes with discontinuous drifts, expanding understanding of jump processes with irregular coefficients.

Contribution

It proves pathwise uniqueness for SDEs with time-dependent Sobolev drifts driven by symmetric alpha-stable processes, allowing for jump discontinuities in the drift.

Findings

Pathwise uniqueness holds for alpha in (1,2).

Drifts with jump discontinuities are permitted when alpha is in (2d/(d+1), 2).

Estimates of Krylov's type are used for purely discontinuous semimartingales.

Abstract

In this article we prove the pathwise uniqueness for stochastic differential equations in $\mR^d$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha$-stable processes provided that $\alpha\in(1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha\in(\frac{2d}{d+1},2)$. Our proof is based on some estimates of Krylov's type for purely discontinuous semimartingales.