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

Marc Hoffmann; Yating Liu

doi:10.5802/smai-jcm.138

¹CEREMADE, CNRS, Université Paris-Dauphine, PSL and Institut Universitaire de France, 75016 Paris, France
²CEREMADE, CNRS, Université Paris-Dauphine, PSL, 75016 Paris, France

The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 103-134

Abstract

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.

Published online: 2026-03-05

DOI: 10.5802/smai-jcm.138

Classification: 60J60, 65C30, 65C35
Keywords: Interacting particle systems, McKean–Vlasov models, Euler scheme, Oracle inequalities, Lepski’s method.

Author's affiliations:

Marc Hoffmann ¹ ; Yating Liu ²

¹ CEREMADE, CNRS, Université Paris-Dauphine, PSL and Institut Universitaire de France, 75016 Paris, France
² CEREMADE, CNRS, Université Paris-Dauphine, PSL, 75016 Paris, France

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     title = {A statistical approach for simulating the density solution of a {McKean{\textendash}Vlasov} equation},
     journal = {The SMAI Journal of computational mathematics},
     pages = {103--134},
     year = {2026},
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Marc Hoffmann; Yating Liu. A statistical approach for simulating the density solution of a McKean–Vlasov equation. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 103-134. doi: 10.5802/smai-jcm.138

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