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

In this paper, we introduce fully implementable, adaptive Euler-Maruyama schemes for McKean-Vlasov SDEs assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity for either. 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 approach shows in most cases superior performance compared to tamed approximation schemes. In addition, we introduce and analyse an adaptive Milstein scheme for a certain sub-class of McKean-Vlasov SDEs with linear measure-dependence of the drift.

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