A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

TL;DR

This paper introduces an explicit splitting method for semi-linear SDEs with polynomially growing drift, demonstrating superior structural property preservation and convergence, with applications to the FitzHugh-Nagumo model.

Contribution

The paper presents a novel explicit splitting method for SDEs with locally Lipschitz drift, ensuring structural preservation and mean-square convergence, outperforming existing Euler-Maruyama variants.

Findings

Method is mean-square convergent of order 1.

Preserves oscillatory dynamics and structural properties.

Effective in statistical inference contexts.

Abstract

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.