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

12/30/2022

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by   Yating Liu, et al.

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In this paper, we study three numerical schemes for the McKean-Vlasov equation dX_t=b(t, X_t, μ_t) dt+σ(t, X_t, μ_t) dB_t, ∀ t∈[0,T], μ_t is the probability distribution of X_t, where X_0 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 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.