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Pré-Publication, Document De Travail Année : 2022

Nonparametric adaptive estimation for interacting particle systems

Fabienne Comte
Valentine Genon-Catalot
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Résumé

We consider a stochastic system (X N i (t), i = 1,. .. , N) of N interacting particles with constant diusion coecient and linear drift b(t, x, µ) = α(t)x − β(t) (x − y)µ(dy) depending on two unknown deterministic functions α(t), β(t). Our concern here is the nonparametric estimation of these functions from a continuous observation of the process on [0, T ] for xed T and large N. We dene two collections of projection estimators αm(t), γ p (t) respectively of α(t), γ(t) = α(t) − β(t) where for each m (resp. p), αm(t) (resp. γ p (t)) belongs to a nite dimensional subspace of L 2 ([0, T ]). We study the L 2-risks of these estimators where the risk is dened either by the expectation of an empirical norm or by the expectation of a deterministic norm. Afterwards, we propose a data-driven choice m (resp. p) of the value m (resp. p) and study the risk of the adaptive estimators. The case of β(t) ≡ 0 is also treated separately. The results are illustrated by numerical experiments on simulated data.
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Dates et versions

hal-03696877 , version 1 (16-06-2022)
hal-03696877 , version 2 (09-05-2023)

Identifiants

  • HAL Id : hal-03696877 , version 2

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Fabienne Comte, Valentine Genon-Catalot. Nonparametric adaptive estimation for interacting particle systems. 2022. ⟨hal-03696877v2⟩
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