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

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  • Fabio Antonelli
  • Arturo Kohatsu

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.

Suggested Citation

  • Fabio Antonelli & Arturo Kohatsu, 1998. "Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation," Economics Working Papers 311, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:311
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    Keywords

    Mc Kean-Vlasov equation; Malliavin calculus;

    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General

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