Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes

We consider the one-dimensional stochastic differential equation equation* Xₜ = x₀ + Lₜ + ₀ᵗ (Xₛ) ds, t 0, equation* where is a finite measure of Kato class K_ with (0, -1] and (Lₜ) ₓ ₀ is a symmetric -stable process with (1, 2). We derive weak and strong well posedness for this equation when -1 and < -1, respectively, and show that the condition -1 is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution (Xₓ) ₓ ₀ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, -stable processes with (1, 2) that is perturbed by an adapted, right-continuous process of finite variation.