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.

中文翻译：

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McKean 方程和相互作用粒子系统的 Milstein 方案的一阶收敛

在本文中，我们为 McKean 随机微分方程 (McKean SDE) 推导出完全可实现的一阶时间步进方案，允许状态分量中具有超线性增长的漂移项。我们为与 McKean 方程相关的时间离散相互作用粒子系统提出了 Milstein 方案，并证明了 1 阶和矩稳定性的强收敛性，如果只有单边 Lipschitz 条件成立，则可以抑制漂移。为了推导出强收敛率的主要结果，我们在具有有限二阶矩的概率测度空间上使用微积分。此外，还提供了支持我们的理论发现的数值例子。