Abstract: We prove pathwise uniqueness for stochastic differential
equations driven by non-degenerate

symmetric $\alpha$-stable L\'evy processes with values in ${\bf R}^d$
having a bounded and  $\beta$-H\"older con-

tinuous drift term. We assume $\beta  > 1 - \alpha/2 $ and  $\alpha
\in  [1; 2)$. The proof requires analytic

regularity results for the associated non-local operators of
Kolmogorov type. We also

study differentiability of solutions with respect to initial
conditions and the homeomorphism

property of solutions.