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Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

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  • Comte, Fabienne
  • Prieur, Clémentine
  • Samson, Adeline

Abstract

The paper considers a process Zt=(Xt,Yt) where Xt is the position of a particle and Yt its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically β-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 δ. 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.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:11:p:3689-3718
    DOI: 10.1016/j.spa.2017.03.011
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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. Asin, Nicolas & Johannes, Jan, 2016. "Adaptive non-parametric estimation in the presence of dependence," LIDAM Discussion Papers ISBA 2016007, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. 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.
    4. 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.
    5. Comte, F. & Genon-Catalot, V. & Rozenholc, Y., 2009. "Nonparametric adaptive estimation for integrated diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 811-834, March.
    6. 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.
    7. Arnaud Gloter, 2006. "Parameter Estimation for a Discretely Observed Integrated Diffusion Process," Post-Print hal-00404901, HAL.
    8. Comte, F. & Merlevède, F., 2005. "Super optimal rates for nonparametric density estimation via projection estimators," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 797-826, May.
    9. Arnaud Gloter, 2006. "Parameter Estimation for a Discretely Observed Integrated Diffusion Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 83-104, March.
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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. 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.

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