Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes

We consider the one-dimensional stochastic differential equation {equation*} X_t = x_0 + L_t + ∫_0^t μ(X_s)ds, t ≥ 0, {equation*} where μ is a finite measure of Kato class Kη with η ∈ (0,α-1] and (Lₜ)t ≥ 0 is a symmetric α-stable process with α ∈ (1,2). We derive weak and strong well posedness for this equation when η ≤α-1 and η < α-1, respectively, and show that the condition η ≤ α-1 is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution (Xₜ)t ≥ 0 and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, α-stable processes with α ∈ (1,2) that is perturbed by an adapted, right-continuous process of finite variation.