First exit times of SDEs driven by stable Lévy processes

Abstract

We study the exit problem of solutions of the stochastic differential equation d X t ε = − U ′ ( X t ε ) d t + ε d L t from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system Y ̇ t = − U ′ ( Y t ) . The process L is composed of a standard Brownian motion and a symmetric α -stable Lévy process. Using probabilistic estimates we show that, in the small noise limit ε → 0 , the exit time of X ε from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the α -stable component of L , the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations.