Supercritical SDEs driven by multiplicative stable-like L\'evy processes

In this paper, we study the following time-dependent stochastic differential equation (SDE) in \mathbb {R}^d: \begin{equation*} \mathrm {d} X_{t}= \sigma (t, X_{t-}) \mathrm {d} Z_t + b(t, X_{t})\mathrm {d} t, \quad X_{0}=x\in \mathbb {R}^d, \end{equation*} where Z is a d-dimensional non-degenerate \alpha-stable-like process with \alpha \in (0,2), and uniform in t\geqslant 0, x\mapsto \sigma (t, x): \mathbb {R}^d\to \mathbb {R}^d\otimes \mathbb {R}^d is \beta-order Hölder continuous and uniformly elliptic with \beta \in ( (1-\alpha )^+ , 1), and x\mapsto b(t, x) is \beta-order Hölder continuous. The Lévy measure of the Lévy process Z can be anisotropic or singular with respect to the Lebesgue measure on \mathbb {R}^d and its support can be a proper subset of \mathbb {R}^d. We show in this paper that for every starting point x \in \mathbb {R}^d, the above SDE has a unique weak solution. We further show that the above SDE has a unique strong solution if x\mapsto \sigma (t, x) is Lipschitz continuous and x\mapsto b(t, x) is \beta-order Hölder continuous with \beta \in (1-\alpha /2,1). When \sigma (t, x)=\mathbb {I}_{d\times d}, the d\times d identity matrix, and Z is an arbitrary non-degenerate \alpha-stable process with 0