Nonparametric inference for density-dependent McKean--Vlasov diffusions
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 (bnlog n/n)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 (bnlog n/n)β/(2β+3) in the L2-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.
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BibTeX
@misc{belomestny2026nonparametric,
title = {{Nonparametric inference for density-dependent McKean--Vlasov diffusions}},
author = {Denis Belomestny and Ekaterina Morozova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01166},
eprint = {2609.01166},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01166},
}
RIS
TY - UNPB
TI - Nonparametric inference for density-dependent McKean--Vlasov diffusions
AU - Denis Belomestny
AU - Ekaterina Morozova
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01166
UR - https://arxiv.org/abs/2609.01166
AB - 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_nlog n/n)^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 (b_nlog n/n)^β/(2β+3) in the L^2-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.
ER -