A multilevel Monte Carlo method for a class of McKean-Vlasov processes

Abstract

We generalize the multilevel Monte Carlo (MLMC) method of Giles to the simulation of systems of particles that interact via a mean field. When the number of particles is large, these systems are described by a McKean-Vlasov process - a stochastic differential equation (SDE) whose coefficients depend on expectations of the solution as well as pathwise data. In contrast to standard MLMC, the new method uses mean field estimates at coarse levels to inform the fine level computations. Using techniques from the theory of propagation of chaos, we prove convergence and complexity results for the algorithm in a special case. We find that the new method achieves $L^1$ error of size $\varepsilon$ with $O(\varepsilon^{-2} (\log \varepsilon)^5)$ complexity, in contrast to the $O(\varepsilon^{-3})$ complexity of standard methods. We also prove a variance scaling result that strongly suggests similar performance of the algorithm in a more general context. We present numerical examples from applications and observe the expected behavior in each case.