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 Hölder constraints. Using the endpoint-adapted graded approximation, we achieve the rate of ${(b_n\log n\mathrm{/}n)}^{2(\beta +1)\mathrm{/}(2\beta +3)}$ for the Kullback-Leibler divergence between the true and estimated stationary densities, with $b_n$ 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 ${(b_n\log n\mathrm{/}n)}^{\beta \mathrm{/}(2\beta +3)}$ in the $L^2$-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.