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Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

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  • Cattiaux, Patrick
  • León, José R.
  • Prieur, Clémentine

Abstract

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.

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  • Cattiaux, Patrick & León, José R. & Prieur, Clémentine, 2014. "Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1236-1260.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:3:p:1236-1260
    DOI: 10.1016/j.spa.2013.10.008
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    References listed on IDEAS

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    1. Samson, Adeline & Thieullen, Michèle, 2012. "A contrast estimator for completely or partially observed hypoelliptic diffusion," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2521-2552.
    2. Yvo Pokern & Andrew M. Stuart & Petter Wiberg, 2009. "Parameter estimation for partially observed hypoelliptic diffusions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 49-73, January.
    3. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
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    Cited by:

    1. Dexheimer, Niklas & Strauch, Claudia, 2022. "Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observations," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 321-362.
    2. Comte, Fabienne & Prieur, Clémentine & Samson, Adeline, 2017. "Adaptive estimation for stochastic damping Hamiltonian systems under partial observation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3689-3718.
    3. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    4. Anna Melnykova, 2020. "Parametric inference for hypoelliptic ergodic diffusions with full observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 595-635, October.
    5. Cattiaux Patrick & León José R. & Prieur Clémentine, 2017. "Invariant density estimation for a reflected diffusion using an Euler scheme," Monte Carlo Methods and Applications, De Gruyter, vol. 23(2), pages 71-88, June.

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