Paper #311

Title:

Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation

Authors:

Fabio Antonelli and Arturo Kohatsu

Date:

August 1998

Abstract:

This paper studies the rate of convergence of an appropriate discretization scheme of the solution of the Mc Kean-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 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 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 and in the sup norm is of the order of $\frac 1{\sqrt n} + h $, while for the densities is of the order $ h +\frac 1 {\sqrt {nh}}$. This result is obtained by carefully employing techniques of Malliavin Calculus.

Keywords:

Mc Kean-Vlasov equation, Malliavin calculus

JEL codes:

C15

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

Annals of Applied Probability, 12, (2002), pp. 423-476

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