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

Abdul-Lateef Haji-Ali, Håkon Hoel, Raúl Tempone

TL;DR

The paper addresses the weak convergence rate of particle approximations to the McKean-Vlasov equation. It develops a framework based on the Kolmogorov backward equation for the particle system and uses stochastic flows together with dual functions to bound the weights in a dual-weighted residual representation, thereby establishing a weak-error rate of $O(n^{-1})$. The main result is proved under Lipschitz conditions on the drift $a$, diffusion $\sigma$, and interaction kernels $\kappa_1,\kappa_2$, together with a priori moment bounds for the MV solution. The approach generalizes prior results that required $\kappa_2\equiv 0$ and provides a pathway to relax assumptions through numerical evidence, with explicit bounds on the empirical kernel deviations and moment growth. Numerical experiments corroborate the rate and suggest that some regularity assumptions may be relaxable.

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.

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

TL;DR

The paper addresses the weak convergence rate of particle approximations to the McKean-Vlasov equation. It develops a framework based on the Kolmogorov backward equation for the particle system and uses stochastic flows together with dual functions to bound the weights in a dual-weighted residual representation, thereby establishing a weak-error rate of . The main result is proved under Lipschitz conditions on the drift , diffusion , and interaction kernels , together with a priori moment bounds for the MV solution. The approach generalizes prior results that required and provides a pathway to relax assumptions through numerical evidence, with explicit bounds on the empirical kernel deviations and moment growth. Numerical experiments corroborate the rate and suggest that some regularity assumptions may be relaxable.

Abstract

We present a proof showing that the weak error of a system of interacting stochastic particles approximating the solution of the McKean-Vlasov equation is . 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.

Paper Structure

This paper contains 2 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. A Bound on the Weak Error

Key Result

Proposition 2.1

Assume that weak solutions to eq:Z-MV exist and let $\{Z_{i}\}_{i=1}^{n}$ be $n$ independent processes each satisfying eq:Z-MV with independent underlying Wiener processes. Let $\kappa : \mathbb{R}^{d} \times \mathbb{R}^{d} \to \mathbb{R}$ be a Lipschitz continuous function, i.e., there exists a con Let $p \in \{1, 2, \ldots\}$, then for any $i \in \{1, \ldots, n\}$, we have Moreover, assuming th

Theorems & Definitions (2)

  • Proposition 2.1
  • proof