Particle method for the numerical simulation of the path-dependent McKean-Vlasov equation

Abstract

We present the particle method for simulating the solution to the path-dependent McKean-Vlasov equation, in which both the drift and the diffusion coefficients depend on the whole trajectory of the process up to the current time t, as well as on the corresponding marginal distributions. Our main contribution is the derivation of explicit convergence rates that capture the interplay between time and space discretization. To control the uniform-in-time convergence of empirical measures in Wasserstein distance on a fixed interval, we develop two approaches: one based on Fournier-Guillin estimates, the other extending Horowitz-Karandikar's method to general p greater or equal to 2. We then compare the respective regimes of applicability. Numerical simulations of a generalized Ornstein-Uhlenbeck process with memory provide evidence for the accuracy of our bounds. We also apply our method to an extension of the Jansen-Rit mean-field model for neural masses.