Discontinuous Stochastic Differential Equations Driven by L\'evy Processes

In this article we prove the pathwise uniqueness for stochastic differential equations in ${\text{mR}}^d$ 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.