Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

The paper considers a process $Z_t=(X_t,Y_t)$ where $X_t$ is the position of a particle and $Y_t$ its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically $\beta$-mixing. In this context, we propose an adaptive non-parametric kernel estimator of the stationary density $p$ of $Z$, based on $n$ discrete time observations with time step $\delta$. Two observation schemes are considered: in the first one, $Z$ is the observed process, in the second one, only $X$ is measured. Estimators are proposed in both settings and upper risk bounds of the mean integrated squared error (MISE) are proved and discussed in each case, the second one being more difficult than the first one. We propose a data driven bandwidth selection procedure based on the Goldenshluger and Lespki (2011) method. In both cases of complete and partial observations, we can prove a bound on the MISE asserting the adaptivity of the estimator. In practice, we take advantage of a very recent improvement of the Goldenshluger and Lespki (2011) method provided by Lacour et al. (2016), which is computationally efficient and easy to calibrate. We obtain convincing simulation results in both observation contexts.