Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation

TL;DR

This paper introduces and analyzes three numerical schemes for the McKean-Vlasov equation, including particle and quantization-based methods, providing convergence rates and practical simulations for complex systems.

Contribution

It extends convergence analysis of particle methods to higher dimensions and introduces two novel quantization-based schemes for McKean-Vlasov equations.

Findings

Proved convergence rate of particle method in Wasserstein distance.

Analyzed recursive quantization scheme for Vlasov setting.

Simulated applications on Burger's equation and neuron network.

Abstract

In this paper, we study three numerical schemes for the McKean-Vlasov equation \[\begin{cases} \;dX_t=b(t, X_t, \mu_t) \, dt+\sigma(t, X_t, \mu_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;\mu_t \text{ is the probability distribution of }X_t, \end{cases}\] where $X_0$ is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients $b$ and $\sigma$, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends a previous work [BT97] established in one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme, inspired by the $K$-means clustering). Two examples are simulated at the end of this paper: Burger's equation and the network of FitzHugh-Nagumo neurons in dimension 3.