Strong existence and uniqueness for stable stochastic differential equations with distributional drift

Abstract

We consider the stochastic differential equation $$d{X}_t=b(X_t)dt+d{L}_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $\alpha$-stable Lévy processes, $\alpha \in (1,2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--Hölder space ${\mathcal{C}}^{\beta }$ for $\beta >1/2-\alpha /2$.