Nonlinear stochastic equations with multiplicative L\'evy noise

TL;DR

This paper solves the Langevin equation with multiplicative Lévy noise, analyzing its solutions, calculus rules, and escape dynamics, revealing algebraic asymptotics and finite variance under certain interpretations.

Contribution

It provides a solution to the Langevin equation with multiplicative Lévy noise and compares different stochastic calculus interpretations.

Findings

Solution exhibits algebraic asymptotic form

Variance can be finite under Stratonovich interpretation

Escape dynamics differ across interpretations

Abstract

The Langevin equation with a multiplicative L\'evy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escaping from a potential well is analysed numerically; predictions of different interpretations of the stochastic integral are compared.