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

Abstract

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= \sigma_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate $\alpha$-stable-like process with $\alpha \in(0,2)$ (including cylindrical case), and uniform in $t\geq 0$, $x\mapsto \sigma_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t (x)$ is $\beta$-order H\"older continuous with $\beta\in(1-\alpha/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x \in {\bf R}^d$. When $\sigma_t (x)={\bf I}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015).