In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs), 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 L\'evy areas.

翻译：本文首先针对与点延迟McKean-Vlasov随机微分方程（McKean-Vlasov SDEs）相关联的相互作用粒子系统推导Milstein格式，其中漂移项可能在状态分量上呈现超线性增长。我们利用具有有限二阶矩的概率测度空间上的变分微积分技术，证明了强收敛阶为1以及矩稳定性。随后，我们引入一种对偶多层Milstein格式，无需模拟Lévy区域即可为延迟McKean-Vlasov方程解的函数期望构建最优复杂度估计量。