First exit times for lévy-driven diffusions with exponentially light jumps

Abstract

We consider a dynamical system describced by the differential equation Yt =-U(Yt) with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity e to obtain the stochastic differential equation dXεt =-U(X εt-)dt + ε dLt . The process L is a symmetric Lévy process whose jump measure v has exponentially light tails, v([u,∞)) ̃ exp(-uα), α > 0, u→∞. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (-1, 1). In the small noise limit ε → 0, the law of the first exit time σx , x (-1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index α = 1, namely, lnEσ ̃ ε-α for 0 < α < 1, whereas lnEs ̃ ε-1|ln ε|1-1/α for α >1. © Institute of Mathematical Statistics, 2009.