How to simulate L\'evy flights in a steep potential: An explicit splitting numerical scheme

TL;DR

This paper introduces an explicit numerical scheme for simulating stochastic differential equations driven by heavy-tailed Le9vy noise in steep potentials, accurately capturing moments and preventing solution explosion.

Contribution

The paper presents a novel explicit splitting scheme specifically designed for SDEs with superlinear drift and Le9vy noise, ensuring stability and moment accuracy.

Findings

Prevents explosion in simulations of Le9vy-driven SDEs

Accurately reproduces moments of solutions in steep potentials

Effective for statistical analysis of Le9vy flights

Abstract

We propose an effective explicit numerical scheme for simulating solutions of stochastic differential equations with confining superlinear drift terms, driven by multiplicative heavy-tailed L\'evy noise. The scheme is designed to prevent explosion and accurately capture all finite moments of the solutions. In the purely Gaussian case, it correctly reproduces moments of sub-Gaussian tails of the solutions. This method is particularly well-suited for approximating statistical moments and other probabilistic characteristics of L\'evy flights in steep potential landscapes.