Rate of Convergence of a Particle Method to the Solution of the Mc Kean - Vlasov's Equation

This paper studies the rate of convergence of an appropriate discretization scheme of the solution of the Mc Kean Vlasov equation introduce d by Bossy and Talay. More speci cally, we consider approximations of the distribution and of the density of the solution of the stochastic di erential equation associated to the Mc Kean 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 st ep in the time discretization, we prove that the rate of convergence of the distribution functions of the approximating sequence in the L( R) norm and in the sup norm is of the order of 1 p n +h, while for the densities is of the o rder h+ 1 p nh . This result is obtained by carefully employing techniques of Malliavin Calculus.