Rate of Estimation for the Stationary Distribution of Stochastic Damping Hamiltonian Systems with Continuous Observations
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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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