Weak convergence analysis in the particle limit of the McKean--Vlasov equations using stochastic flows of particle systems

TL;DR

This paper proves that the weak error of particle systems approximating McKean-Vlasov equations decreases at a rate of 1/n, using stochastic flow analysis and derivative bounds, with numerical verification.

Contribution

It introduces a novel proof technique based on the Kolmogorov backward equation and derivative bounds for analyzing particle system convergence.

Findings

Weak error is of order 1/n.

Numerical experiments confirm theoretical convergence rate.

Assumptions can be relaxed as indicated by experiments.

Abstract

We present a proof showing that the weak error of a system of $n$ interacting stochastic particles approximating the solution of the McKean-Vlasov equation is $\mathcal O(n^{-1})$. Our proof is based on the Kolmogorov backward equation for the particle system and bounds on the derivatives of its solution, which we derive more generally using the variations of the stochastic particle system. The convergence rate is verified by numerical experiments, which also indicate that the assumptions made here and in the literature can be relaxed.