Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

Abstract

The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian $\alpha$-stable type Lévy motions. An efficient and accurate numerical scheme is developed and validated for computing the mean exit time and escape probability from the governing differential-integral equation. An asymptotic solution for the mean exit time is given when the pure jump measure in the Lévy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of $\alpha$ in the $\alpha$-stable Lévy jump measure. The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the value of $\alpha$ is in (0,1).