Pathwise uniqueness for singular SDEs driven by stable processes

TL;DR

This paper establishes pathwise uniqueness for certain stochastic differential equations driven by symmetric stable processes with irregular drift, using analytic regularity of related integro-differential operators.

Contribution

It proves pathwise uniqueness under minimal regularity assumptions on the drift for SDEs driven by stable processes, extending previous results.

Findings

Pathwise uniqueness holds for SDEs with Hölder continuous drift.

Solutions are differentiable with respect to initial conditions.

Solutions form a homeomorphism under the given conditions.

Abstract

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $\R^d$ having a bounded and $\beta$-H\"older continuous drift term. We assume $\beta > 1 - \frac{\alpha}{2} $ and $\alpha \in [ 1, 2)$. The proof requires analytic regularity results for associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.