In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean–Vlasov stochastic differential equations, possibly with a drift term exhibiting super-linear growth in the state component. We prove strong convergence of order one and moment stability, making use of techniques from variational calculus on the space of probability measures with finite second-order moments. Then, we introduce an antithetic multilevel Milstein scheme, which leads to optimal complexity estimators for expected functionals of solutions to delay McKean–Vlasov equations without the need to simulate Lévy areas.

中文翻译：

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延迟 McKean-Vlasov 方程和相互作用粒子系统的 Milstein 方案和对立多级蒙特卡罗采样

在本文中，我们首先推导出与点延迟 McKean-Vlasov 随机微分方程相关的相互作用粒子系统的 Milstein 方案，可能具有在状态分量中表现出超线性增长的漂移项。我们利用有限二阶矩的概率测度空间上的变分微积分技术，证明了一阶和矩稳定性的强收敛性。然后，我们引入了一种对立的多级 Milstein 方案，该方案可以为解的预期函数提供最优复杂度估计器，从而延迟 McKean-Vlasov 方程，而无需模拟 Lévy 区域。