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

摘要

We address the approximation of functionals depending on a system of particles, describedby stochastic differential equations (SDEs), in the mean-field limit when the number of particlesapproaches infinity. This problem is equivalent to estimating the weak solution of the limitingMcKean-Vlasov SDE. To that end, our approach uses systems with finite numbers of particlesand a time-stepping scheme. In this case, there are two discretization parameters: the numberof time steps and the number of particles. Based on these two parameters, we consider differentvariants 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 onetypical setting with an error tolerance of TOL, is O (TOL-3)when using the partitioningestimator and the Milstein time-stepping scheme. We also consider a method that uses therecent Multi-index Monte Carlo method and show an improved work complexity in the sametypical setting of O(TOL-2 log(TOL-1)2). Our numerical experiments are carried out on theso-called Kuramoto model, a system of coupled oscillators.