Discontinuous Stochastic Differential Equations Driven by L\'evy Processes

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.