Pathwise uniqueness for singular SDEs driven by stable processes

Abstract

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable Lévy processes with values in ${\mathbb{R}}^d$ having a bounded and $\beta$-Hölder 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.

The main change is the new statement (iii) in Theorem 1.1 about differentiability of solutions with respect to initial conditions