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.

中文翻译：

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McKean-Vlasov 方程和相互作用粒子系统的 Milstein 方案的一阶收敛

在本文中，我们推导出完全可实现的 McKean-Vlasov 随机微分方程的一阶时间步进方案，允许状态分量具有超线性增长的漂移项。我们为与 McKean-Vlasov 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。