On the α-dependence of stochastic differential equations with Hölder drift and driven by α-stable Lévy processes

Abstract

In this study, we investigate the convergence behavior as α→2 of the solutions to stochastic differential equations (SDEs) with Lipschitz and Hölder drifts, and driven by α-stable Lévy processes. First, we prove that the solution of an SDE driven by an α-stable Lévy process converges weakly to that of an SDE forced by Brownian motion in the Skorokhod space. We mainly employ Aldous' criterion and martingale characterization. Then, by using an abstract result given by Gyöngy and Krylov, we prove that the solution of an SDE driven by the subordinated Brownian motion via an α/2-stable subordinator strongly converges to that of an SDE forced by Brownian motion. Based on uniform estimates of the mean exit times and related probabilistic representations, we also present an α-dependent result for fractional diffusion equations. Finally, given α⁎ in (1,2), we discuss the α-continuity of the solution to an SDE driven by an α-stable Lévy process at α⁎.