A statistical approach for simulating the density solution of a McKean–Vlasov equation

We prove optimal convergence results of a stochastic particle method for computing the classical solution of a multivariate McKean–Vlasov equation, when the measure variable is in the drift, following the classical approach of [1, 11]. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel type estimator. In particular, we generalise the Bernstein inequality of [18] for mean-field McKean–Vlasov models to interacting particles Euler schemes and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study, and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters.