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

Abstract

We consider the stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dL_{t},\end{equation*} 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 $\beta >1/2-\alpha /2$.