Simulation of Non-Lipschitz Stochastic Differential Equations Driven by $\alpha$-Stable Noise: A Method Based on Deterministic Homogenization
Georg A. Gottwald, Ian Melbourne · Multiscale Modeling and Simulation · 2021
We devise an explicit method to integrate $\alpha$-stable stochastic differential equations (SDEs) with nonglobally-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the Lévy noise, we use a deterministic map which has the desired SDE as its homogenized 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 discretization fails to do so. As a by-product we devise an entirely deterministic method to construct $\alpha$-stable laws.