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

2020/05/12 by Christoph Reisinger, Reisinger, Christoph, Wolfgang Stockinger +1 · 5 citations
Economics, Econometrics and Finance · Physics and Astronomy · Social Sciences · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2005.06034

openalex publication_date 2020/05/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasov\nstochastic differential equations (SDEs) assuming only a standard monotonicity\ncondition on the drift and diffusion coefficients but no global Lipschitz\ncontinuity in the state variable for either, while global Lipschitz continuity\nis required for the measure component only. We prove moment stability of the\ndiscretised processes and a strong convergence rate of 1/2. Several numerical\nexamples, centred around a mean-field model for FitzHugh-Nagumo neurons,\nillustrate that the standard uniform scheme fails and that the adaptive\napproach shows in most cases superior performance to tamed approximation\nschemes. In addition, we introduce and analyse an adaptive Milstein scheme for\na certain sub-class of McKean-Vlasov SDEs with linear measure-dependence of the\ndrift.\n

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• Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity
• Strong error analysis and first-order convergence of Milstein-type schemes for McKean-Vlasov SDEs with superlinear coefficients
• Euler-type methods for Levy-driven McKean-Vlasov SDEs with super-linear coefficients: mean-square error analysis
• Long time strong convergence analysis of one-step methods for McKean-Vlasov SDEs with superlinear growth coefficients
• The adaptive EM schemes for McKean-Vlasov SDEs with common noise in finite and infinite horizons

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