First exit times of solutions of stochastic differential equations driven by multiplicative Levy noise with heavy tails

TL;DR

This paper investigates the asymptotic behavior of the first exit times from bounded domains for gradient systems perturbed by small multiplicative Levy noise, analyzing different stochastic calculus interpretations.

Contribution

It determines the asymptotics of first exit times for solutions of SDEs driven by heavy-tailed Levy noise under various stochastic calculus frameworks.

Findings

Asymptotic formulas for first exit times are derived.

Differences between Itô, Stratonovich, and Marcus interpretations are clarified.

Heavy-tailed Levy noise significantly influences exit time behavior.

Abstract

In this paper we study first exit times from a bounded domain of a gradient dynamical system $\dot Y_t=-\nabla U(Y_t)$ perturbed by a small multiplicative L\'evy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular we determine the asymptotics of the first exit time of solutions of It\^o, Stratonovich and Marcus canonical SDEs.