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

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.