Abstract (EN)

In this paper, we study three numerical schemes for the McKean-Vlasov equation dXt = b(t, Xt, µt) dt + σ(t, Xt, µt) dBt, ∀ t ∈ [0, T ], µt is the probability distribution of Xt, where X0 : (Ω, F, P) → (R^d , B(R^d)) is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients b and σ, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends 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 introduced in [BT97] and the network of FitzHugh-Nagumo neurons (see [BFFT12] and [BFT15]) in dimension 3.