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

TL;DR

This paper analyzes the exit times of solutions to non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components, revealing exponential distribution behavior in the small noise limit and highlighting differences from Gaussian perturbations.

Contribution

It provides a probabilistic analysis of exit times for SDEs with alpha-stable Levy noise, extending classical results to heavy-tailed jump processes.

Findings

Exit times are exponentially distributed as noise vanishes.

Expected exit times depend on the heavy-tail properties of the Levy process.

Results differ significantly from Gaussian-driven systems.

Abstract

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.