On pathwise uniqueness for stochastic differential equations driven by stable Levy processes

We study a one-dimensional stochastic differential equation driven by a stable Levy process of order alpha with drift and diffusion coefficients b, sigma. When alpha is an element of (1, 2), we investigate pathwise uniqueness for this equation. When alpha is an element of (0, 1), we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether alpha is an element of (0, 1) or alpha is an element of (1, 2) and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of b and sigma.