Articlegreen2021

First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems

Jianhai Bao · Christoph Reisinger · Panpan Ren · Wolfgang Stockinger

40 citations0/0 files storedProceedings of the Royal Society A Mathematical Physical and Engineering Sciences

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Stochastic processes and financial applications Stochastic processes and statistical mechanics Numerical methods for differential equations

Abstract

In this paper, we derive fully implementable first-order time-stepping schemes for McKean-Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean-Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

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References

2016Numerical solution of stochastic differential equations 2007A Concise Course on Stochastic Partial Differential Equations 2018Probabilistic Theory of Mean Field Games with Applications I 2010Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients 2012Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients 2017Mean-field stochastic differential equations and associated PDEs 1997A stochastic particle method for the McKean-Vlasov and the Burgers equation 2013A note on tamed Euler approximations

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2021An adaptive Euler–Maruyama scheme for McKean–Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh–Nagumo model 2021On explicit Milstein-type scheme for McKean–Vlasov stochastic differential equations with super-linear drift coefficient

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