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

Abstract

In this article, we construct and analyse explicit numerical splitting methods for a class of semilinear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The methods are proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. In particular, first, they are hypoelliptic in every iteration step. Second, they are geometrically ergodic and have asymptotically bounded second moments. Third, they preserve 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 methods to preserve the aforementioned properties makes them 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.

Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations

“…This corresponds to the case m B = 1, that is all the components of V t have a direct interaction with at least one component of U t and stochasticity only acts on the unobserved variables. This corresponds to the 1-step hypoellipticity defined by Buckwar et al (2021).…”

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

“…This flexibility is understood in the following way: given an MV-SDE it is left to the user to choose which terms form v and which terms form b in (1.1) (within restrictions). Moreover, since there is a split v Vs b the user may decide to add & subtract convenient terms to the drift (see Section 3.4.2 below; also [14,Eqs. (37) and ( 38)]).…”