First order convergence of Milstein schemes for McKean equations and interacting particle systems
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In this paper, we derive fully implementable first order time-stepping schemes for McKean stochastic differential equations (McKean SDEs), allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretised interacting particle system associated with the McKean 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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