Nonparametric inference for density-dependent McKean--Vlasov diffusions

The present research is devoted to the nonparametric estimation of a density-dependent drift coefficient in a multivariate McKean--Vlasov diffusion from independent observations at a common time, as well as the stationary density. Under certain assumptions on the (known) potential, we reduce the problem to the one-dimensional one and construct a sieve maximum-likelihood estimator based on sparse ReQU neural networks subject to structural and H"older constraints. Using the endpoint-adapted graded approximation, we achieve the rate of left(bnlog n/nright)2(β+1)/(2β+3) for the Kullback-Leibler divergence between the true and estimated stationary densities, with bn being at most a logarithmic factor. Similarly, it is shown that the constructed estimator for the drift coefficient converges to the true one at the rate of left(bnlog n/nright)β/(2β+3) in the L2-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.