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.