Well-posedness of supercritical SDE driven by L\'evy processes with irregular drifts

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\mathbf{R}}^d$: $$d{X}_t={\sigma }_t(X_{t-})d{Z}_t+b_t(X_t)dt,X_0=x\in {\mathbf{R}}^d,$$ where $Z$ is a $d$-dimensioanl nondegenerate $\alpha$-stable-like process with $\alpha \in (0,2)$ (including cylindrical case), and uniform in $t\ge 0$, $x\mapsto \sigma_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$isLipchitzanduniformlyellipticand$x\mapsto b_t (x)$is$\beta$-orderH\ddot{o}ldercontinuouswith$\beta\in(1-\alpha/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x\in {\mathbf{R}}^d$. When ${\sigma }_t(x)={\mathbf{I}}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015).