How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme
Ilya Pavlyukevich, Olga Aryasova, Alexei Chechkin, Oleksii Kulyk
TL;DR
This work addresses numerical simulation of Lévy-driven SDEs with superlinear drift in steep confining potentials, where classical Euler methods can explode or misrepresent moments. It introduces a direct splitting scheme that decouples the deterministic drift flow from the stochastic jump part, ensuring boundedness and exact-like preservation of moments up to order just below the tail index (e.g., q < 9 for Cauchy noise). An error analysis on linear systems demonstrates superior stability and moment accuracy over Euler and reverse splitting, especially in stiff regimes, and the general formulation extends to multi-dimensional settings with convergence guarantees. The paper also presents ready-to-use examples with explicit drift flows, enabling practical implementation across a range of nonlinear, heavy-tailed systems, and provides rigorous conditions under which convergence holds. Overall, the direct splitting approach delivers stable, tail-preserving simulations of Lévy flights in steep potentials with broad applicability and improved reliability for moment and autocorrelation estimation.
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 Lévy 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 Lévy flights in steep potential landscapes.