First exit times of solutions of non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components

Imkeller, Peter; Pavlyukevich, Ilya

First exit times of solutions of non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components

Authors

Description

We study the exit problem of solutions of the stochastic differential equation dX(t)=-U'(X(t))dt+epsilon dL(t) from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system dY=-U'(Y) dt. The process L is composed of a standard Brownian motion and a symmetric alpha-stable Levy process. Using probabilistic estimates we show that in the small noise limit epsilon->0, the exit time of X from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the alpha-stable component of L, the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations.
Slightly revised version. To appear in Stochastic Processes and their Applications. 24 pages, 2 figures