An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh-Nagumo model

Abstract

In this paper, we introduce a fully implementable, adaptive Euler-Maruyama scheme for McKean SDEs with non-globally Lipschitz continuous drifts. We prove moment stability of the discretised processes and a strong convergence rate of 1/2. We present several numerical examples centred around a mean-field model for FitzHugh-Nagumo neurons, which illustrate that the standard uniform scheme fails and that the adaptive scheme shows in most cases superior performance compared to tamed approximation schemes. In addition, we propose a tamed and an adaptive Milstein scheme for a certain class of McKean SDEs.

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

“…To the best of our knowledge this has not yet been discussed for MV-SDE schemes in general. The stability of the SSM provides a theoretical foundation for carrying out simulation with larger timestep and we point to positive results by way of numerical simulation with the Cucker-Smale flocking model [22] where the SSM outperforms both taming [18] and adaptive time-stepping [45] algorithms.…”

“…We prove its convergence and recover the 1/2-convergence rate in rMSE under the same general assumptions as the tamed [18] or the adaptive time-stepping method [45]. No differentability or nondegeneracy assumptions are imposed and stopping-time arguments are fully avoided.…”

“…Namely, one parallelizes the task of solving N -times an R d -system in opposition to the nonparallelizable task of solving the R dN -system. The computational gains of the parallel implementations for the SMM are shown to be on par of those for the explicit taming [18] and adaptive time-stepping [45] algorithms.…”

“…In this work, we focus on the class of MV-SDE with drifts of super-linear growth in their spatial components [4,18,19,33,45], encapsulated in the function v, where b, σ are uniformly Lipschitz (in space and measure), and σ is non-constant. This class of MV-SDEs appears in several practical models in science, for example, in neuroscience [3,12] introduce the mean-field FitzHugh-Nagumo model for a neurons networks in the brain; [11] discuss individual-based and swarming Cucker-Smale interaction models; and models of battery electrodes [21,26].…”

“…In [34] it is proposed a tamed scheme (and a new wellposedness result) for MV-SDE featuring common noise in the particle system (which (1.2) does not) and where the diffusion is also allowed to grow super-linearly. Adaptive time-stepping methods come as an alternative to taming and in the context of MV-SDE they are proposed in [45].…”