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

Abstract

We consider a dynamical system described by the differential equation ${\dot{Y}}_t=-U^{\prime }(Y_t)$ with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity $\epsilon$ to obtain the stochastic differential equation $d{X}_t^{\epsilon }=-U^{\prime }(X_{t-}^{\epsilon })dt+\epsilon d{L}_t.$ The process $L$ is a symmetric Lévy process whose jump measure $\nu$ has exponentially light tails, $\nu ([u,\infty ))\sim \exp (-u^{\alpha })$, $\alpha >0$, $u\to \infty$. 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 $\epsilon \to 0$, the law of the first exit time ${\sigma }_x$, $x\in (-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $\alpha =1$, namely, $\ln \mathbf{E}\sigma \sim {\epsilon }^{-\alpha }$ for $0<\alpha <1$, whereas $\ln \mathbf{E}\sigma \sim {\epsilon }^{-1}|\ln \epsilon {|}^{1-1/\alpha }$ for $\alpha >1$.

Published in at http://dx.doi.org/10.1214/08-AOP412 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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