RATE OF ESTIMATION FOR THE STATIONARY DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTINUOUS OBSERVATIONS - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Année : 2022

RATE OF ESTIMATION FOR THE STATIONARY DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTINUOUS OBSERVATIONS

Résumé

We study the problem of the non-parametric estimation for the density π of the stationary distribution of a stochastic two-dimensional damping Hamiltonian system (Z_t) t∈[0,T ] = (X_t, Y_t) t∈[0,T ]. From the continuous observation of the sampling path on [0, T ], we study the rate of estimation for π(x_0 , y_0) as T → ∞. We show that kernel based estimators can achieve the rate T^{−v} for some explicit exponent v ∈ (0, 1/2). One finding is that the rate of estimation depends on the smoothness of π and is completely different with the rate appearing in the standard i.i.d. setting or in the case of two-dimensional non degenerate diffusion processes. Especially, this rate depends also on y 0. Moreover, we obtain a minimax lower bound on the L 2-risk for pointwise estimation, with the same rate T^{−v}, up to log(T) terms.
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Dates et versions

hal-02455744 , version 1 (26-01-2020)

Identifiants

Citer

Sylvain Delattre, Arnaud Gloter, Nakahiro Yoshida. RATE OF ESTIMATION FOR THE STATIONARY DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTINUOUS OBSERVATIONS. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2022, 58 (4), pp.1998--2028. ⟨10.1214/21-aihp1237⟩. ⟨hal-02455744⟩
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