Supercritical SDEs driven by multiplicative stable-like Lévy processes

Abstract:

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 $0<\alpha <1$, our strong well-posedness result in particular gives an affirmative answer to the open problem in a paper by Priola.