Dimension-independent convergence rate of propagation of chaos and numerical analysis for McKean-Vlasov stochastic differential equations with coefficients nonlinearly dependent on measure

TL;DR

This paper proves a dimension-independent convergence rate for particle methods approximating McKean-Vlasov SDEs with nonlinear measure dependence, and verifies the results through numerical experiments.

Contribution

It establishes a novel dimension-independent convergence rate for propagation of chaos in nonlinear measure-dependent MV-SDEs and provides numerical analysis for discretization.

Findings

Dimension-independent convergence rate proved for nonlinear measure-dependent MV-SDEs

Numerical experiments confirm theoretical convergence rates

Method extends understanding of particle approximation in high dimensions

Abstract

In contrast to ordinary stochastic differential equations (SDEs), the numerical simulation of McKean-Vlasov stochastic differential equations (MV-SDEs) requires approximating the distribution law first. Based on the theory of propagation of chaos, particle approximation method is widely used. Then, a natural question is to investigate the convergence rate of the method (also referred to as the convergence rate of PoC). In fact, the PoC convergence rate is well understood for MV-SDEs with coefficients linearly dependent on the measure, but the rate deteriorates with dimension $d$ under the $L^p$-Wasserstein metric for nonlinear measure-dependent coefficients, even when Lipschitz continuity with respect to the measure is assumed. The main objective of this paper is to establish a dimension-independent convergence result of PoC for MV-SDEs whose coefficients are nonlinear with respect to the measure component but Lipschitz continuous. As a complement we further give the time discretization of the equations and thus verify the convergence rate of PoC using numerical experiments.