Nonparametric inference for density-dependent McKean--Vlasov diffusions

• URL: http://arxiv.org/abs/2609.01166v1
• Date: Tue, 01 Sep 2026 12:44:35 GMT
• Title: Nonparametric inference for density-dependent McKean--Vlasov diffusions
• Abstract summary: We construct a sieve maximum-likelihood estimator based on sparse ReQU neural networks.<n>It converges to the true one at the rate of $left(b_nlog n/nright)/(2+3)$ in the $L2$-metric.<n>A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.
• Score: 3.094107483966034
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\left(b_n\log n/n\right)^{2(β+1)/(2β+3)}$ 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 $\left(b_n\log n/n\right)^{β/(2β+3)}$ in the $L^2$-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.

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