Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes

Abstract

We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t \mu(X_s)ds, \quad t \geq 0, \end{equation*} where $\mu$ is a finite measure of Kato class $K_{\eta}$ with $\eta \in (0,\alpha-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $\alpha$-stable process with $\alpha \in (1,2)$. We derive weak and strong well posedness for this equation when $\eta \leq\alpha-1$ and $\eta < \alpha-1$, respectively, and show that the condition $\eta \leq \alpha-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $\alpha$-stable processes with $\alpha \in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation.