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

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.