Simulation of non-Lipschitz stochastic differential equations driven by $\alpha$-stable noise: a method based on deterministic homogenisation

TL;DR

This paper introduces a novel explicit numerical method for simulating non-Lipschitz $ ext{α}$-stable SDEs using deterministic homogenisation, effectively handling boundary conditions and overcoming limitations of traditional schemes.

Contribution

The authors develop a deterministic homogenisation-based method for $ ext{α}$-stable SDEs that improves stability and boundary behavior over existing numerical approaches.

Findings

Method respects natural boundaries in simulations.

Overcomes numerical instabilities caused by unbounded Lévy noise.

Provides a deterministic approach to construct $ ext{α}$-stable laws.

Abstract

We devise an explicit method to integrate $\alpha$-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the L\'evy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct $\alpha$-stable laws.