IDEAS home Printed from https://ideas.repec.org/a/bla/jorssb/v81y2019i2p361-384.html
   My bibliography  Save this article

Hypoelliptic diffusions: filtering and inference from complete and partial observations

Author

Listed:
  • Susanne Ditlevsen
  • Adeline Samson

Abstract

The statistical problem of parameter estimation in partially observed hypoelliptic diffusion processes is naturally occurring in many applications. However, because of the noise structure, where the noise components of the different co‐ordinates of the multi‐dimensional process operate on different timescales, standard inference tools are ill conditioned. We propose to use a higher order scheme to approximate the likelihood, such that the different timescales are appropriately accounted for. We show consistency and asymptotic normality with non‐typical convergence rates. When only partial observations are available, we embed the approximation in a filtering algorithm for the unobserved co‐ordinates and use this as a building block in a stochastic approximation expectation–maximization algorithm. We illustrate on simulated data from three models: the harmonic oscillator, the FitzHugh–Nagumo model used to model membrane potential evolution in neuroscience and the synaptic inhibition and excitation model used for determination of neuronal synaptic input.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:81:y:2019:i:2:p:361-384
    DOI: 10.1111/rssb.12307
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/rssb.12307
    Download Restriction: no
    File URL: https://libkey.io/10.1111/rssb.12307?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    2. 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.
    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, 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.
    6. Joshua H Goldwyn & Eric Shea-Brown, 2011. "The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations," PLOS Computational Biology, Public Library of Science, vol. 7(11), pages 1-9, November.
    7. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    8. 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.
    9. Arnaud Gloter, 2006. "Parameter Estimation for a Discretely Observed Integrated Diffusion Process," Post-Print hal-00404901, HAL.
    10. 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.
    11. Susanne Ditlevsen & Michael Sørensen, 2004. "Inference for Observations of Integrated Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(3), pages 417-429, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Matthew M. Graham & Alexandre H. Thiery & Alexandros Beskos, 2022. "Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1229-1256, September.
    3. Quentin Clairon & Adeline Samson, 2022. "Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations," Computational Statistics, Springer, vol. 37(5), pages 2471-2491, November.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Quentin Clairon & Adeline Samson, 2022. "Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations," Computational Statistics, Springer, vol. 37(5), pages 2471-2491, November.
    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. Quentin Clairon & Adeline Samson, 2020. "Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 105-127, April.
    4. 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.
    5. 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.
    6. Shu, Huisheng & Jiang, Ziwei & Zhang, Xuekang, 2023. "Parameter estimation for integrated Ornstein–Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 199(C).
    7. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465, September.
    8. 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.
    9. Nicolau, João, 2008. "Modeling financial time series through second-order stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2700-2704, November.
    10. Yunyan Wang & Lixin Zhang & Mingtian Tang, 2012. "Re-weighted functional estimation of second-order diffusion processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1129-1151, November.
    11. 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.
    12. Song Yuping & Hou Weijie & Zhou Shengyi, 2019. "Variance reduction estimation for return models with jumps using gamma asymmetric kernels," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 23(5), pages 1-38, December.
    13. 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.
    14. Jean Jacod & Mark Podolskij, 2012. "A Test for the Rank of the Volatility Process: The Random Perturbation Approach," Global COE Hi-Stat Discussion Paper Series gd12-268, Institute of Economic Research, Hitotsubashi University.
    15. Jean Jacod & Mark Podolskij, 2012. "A test for the rank of the volatility process: the random perturbation approach," CREATES Research Papers 2012-57, Department of Economics and Business Economics, Aarhus University.
    16. Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    17. Song, Renming & Xie, Longjie, 2020. "Well-posedness and long time behavior of singular Langevin stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1879-1896.
    18. Holbach, Simon, 2020. "Positive Harris recurrence for degenerate diffusions with internal variables and randomly perturbed time-periodic input," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6965-7003.
    19. 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.
    20. Arnaud Gloter, 2007. "Efficient estimation of drift parameters in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(4), pages 495-519, October.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:jorssb:v:81:y:2019:i:2:p:361-384. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/rssssea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.