First order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems

In this paper, we derive fully implementable first order time-stepping schemes for McKean–Vlasov
stochastic differential equations (McKean–Vlasov 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–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.