IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v123y2013i2p603-628.html
   My bibliography  Save this article

Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs

Author

Listed:
  • Pokern, Y.
  • Stuart, A.M.
  • van Zanten, J.H.

Abstract

We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with the precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.

Suggested Citation

  • Pokern, Y. & Stuart, A.M. & van Zanten, J.H., 2013. "Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 603-628.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:603-628
    DOI: 10.1016/j.spa.2012.08.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414912001779
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2012.08.010?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Federico M. Bandi & Peter C. B. Phillips, 2003. "Fully Nonparametric Estimation of Scalar Diffusion Models," Econometrica, Econometric Society, vol. 71(1), pages 241-283, January.
    2. Omiros Papaspiliopoulos & Yvo Pokern & Gareth O. Roberts & Andrew M. Stuart, 2012. "Nonparametric estimation of diffusions: a differential equations approach," Biometrika, Biometrika Trust, vol. 99(3), pages 511-531.
    3. Harry Van Zanten, 2003. "On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 6(2), pages 199-213, May.
    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. Fabian Dunker & Thorsten Hohage, 2014. "On parameter identification in stochastic differential equations by penalized maximum likelihood," Papers 1404.0651, arXiv.org.
    2. Frank Meulen & Moritz Schauer & Jan Waaij, 2018. "Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 603-628, October.
    3. van der Meulen, Frank & Schauer, Moritz & van Zanten, Harry, 2014. "Reversible jump MCMC for nonparametric drift estimation for diffusion processes," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 615-632.

    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. Kim, Jihyun & Park, Joon Y., 2017. "Asymptotics for recurrent diffusions with application to high frequency regression," Journal of Econometrics, Elsevier, vol. 196(1), pages 37-54.
    2. Hjalmarsson, Erik, 2003. "Does the Black-Scholes formula work for electricity markets? A nonparametric approach," Working Papers in Economics 101, University of Gothenburg, Department of Economics.
    3. Bonsoo Koo & Oliver Linton, 2010. "Semiparametric Estimation of Locally Stationary Diffusion Models," STICERD - Econometrics Paper Series 551, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. Ang, Andrew & Kristensen, Dennis, 2012. "Testing conditional factor models," Journal of Financial Economics, Elsevier, vol. 106(1), pages 132-156.
    5. Todorov, Viktor & Zhang, Yang, 2023. "Bias reduction in spot volatility estimation from options," Journal of Econometrics, Elsevier, vol. 234(1), pages 53-81.
    6. Chen, Bin & Song, Zhaogang, 2013. "Testing whether the underlying continuous-time process follows a diffusion: An infinitesimal operator-based approach," Journal of Econometrics, Elsevier, vol. 173(1), pages 83-107.
    7. Yu, Jun, 2012. "Bias in the estimation of the mean reversion parameter in continuous time models," Journal of Econometrics, Elsevier, vol. 169(1), pages 114-122.
    8. Aït-Sahalia, Yacine & Park, Joon Y., 2016. "Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models," Journal of Econometrics, Elsevier, vol. 192(1), pages 119-138.
    9. Federico M. Bandi & Roberto Reno, 2009. "Nonparametric Stochastic Volatility," Global COE Hi-Stat Discussion Paper Series gd08-035, Institute of Economic Research, Hitotsubashi University.
    10. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
    11. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    12. Renò, Roberto, 2008. "Nonparametric Estimation Of The Diffusion Coefficient Of Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1174-1206, October.
    13. Wang, Qiying & Phillips, Peter C.B., 2009. "Asymptotic Theory For Local Time Density Estimation And Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 25(3), pages 710-738, June.
    14. Yuping Song & Weijie Hou & Zhengyan Lin, 2022. "Double Smoothed Volatility Estimation of Potentially Non‐stationary Jump‐diffusion Model of Shibor," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 53-82, January.
    15. Yamamura, Mariko & Shoji, Isao, 2010. "A nonparametric method of multi-step ahead forecasting in diffusion processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(12), pages 2408-2415.
    16. Wooyong Lee & Priscilla E. Greenwood & Nancy Heckman & Wolfgang Wefelmeyer, 2017. "Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise," Statistical Inference for Stochastic Processes, Springer, vol. 20(2), pages 237-252, July.
    17. Kristensen, Dennis, 2008. "Estimation of partial differential equations with applications in finance," Journal of Econometrics, Elsevier, vol. 144(2), pages 392-408, June.
    18. Stan Hurn & J.Jeisman & K.A. Lindsay, 2006. "Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations. Working paper #2," NCER Working Paper Series 2, National Centre for Econometric Research.
    19. Bandi, Federico M. & Renò, Roberto, 2012. "Time-varying leverage effects," Journal of Econometrics, Elsevier, vol. 169(1), pages 94-113.
    20. Aït-Sahalia, Yacine & Park, Joon Y., 2012. "Stationarity-based specification tests for diffusions when the process is nonstationary," Journal of Econometrics, Elsevier, vol. 169(2), pages 279-292.

    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:eee:spapps:v:123:y:2013:i:2:p:603-628. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.