Rate of convergence of a particle method to the solution of the McKean--Vlasov equation

Abstract

This paper studies the rate of convergence of an appropriate discretization scheme of the solution of the McKean--Vlasov equation introduced by Bossy and Talay. More specifically, we consider approximations of the distribution and of the density of the solution of the stochastic differential equation associated to the McKean--Vlasov equation. The scheme adopted here is a mixed one: Euler--weakly interacting particle system. If $n$ is the number of weakly interacting particles and $h$ is the uniform step in the time discretization, we prove that the rate of convergence of the distribution functions of the approximating sequence in the $L^1(\Omega\times \Bbb{R})$ norm is of the order of $\frac{1}{\sqrt n} + h $, while for the densities is of the order $ h +\frac{1}{\sqrt {n}h^{1/4}}$. The rates of convergence with respect to the supremum norm are also calculated. This result is obtained by carefully employing techniques of Malliavin calculus.