Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation

Abstract: We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean-Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of $\mathrm{TOL}$, is $\mathcal O\left({\mathrm{TOL}^{-3}}\right)$ when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of $\mathcal O\left(\mathrm{TOL}^{-2}\log(\mathrm{TOL}^{-1})^2\right)$. Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.