PATHWISE UNIQUENESS FOR SINGULAR SDEs DRIVEN BY STABLE PROCESSES

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 $\beta$-Holder continuous drift term. We assume $\displaystyle{ \beta > 1 - {\alpha}/{2} }$ and $\alpha \in [ 1, 2)$. The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.