Milstein schemes and antithetic multi-level Monte-Carlo sampling for delay McKean-Vlasov equations and interacting particle systems

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 multi-level 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.