IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v129y2019i8p2880-2902.html
   My bibliography  Save this article

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

Author

Listed:
  • Liu, Yanghui
  • Nualart, Eulalia
  • Tindel, Samy

Abstract

We consider a stochastic differential equation with additive fractional noise with Hurst parameter H>1∕2, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate τ as τ→∞, when the solution is observed continuously on the time interval [0,τ]. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyze the particular case of the fractional Ornstein–Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.

Suggested Citation

  • Liu, Yanghui & Nualart, Eulalia & Tindel, Samy, 2019. "LAN property for stochastic differential equations with additive fractional noise and continuous time observation," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2880-2902.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2880-2902
    DOI: 10.1016/j.spa.2018.08.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918304198
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2018.08.008?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. Nualart, David & Ouknine, Youssef, 2002. "Regularization of differential equations by fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 103-116, November.
    2. Alexandre Brouste & Stefano Iacus, 2013. "Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package," Computational Statistics, Springer, vol. 28(4), pages 1529-1547, August.
    3. Alexandre Brouste & Marina Kleptsyna, 2010. "Asymptotic properties of MLE for partially observed fractional diffusion system," Statistical Inference for Stochastic Processes, Springer, vol. 13(1), pages 1-13, April.
    4. M.L. Kleptsyna & A. Le Breton, 2002. "Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process," Statistical Inference for Stochastic Processes, Springer, vol. 5(3), pages 229-248, October.
    5. Le Breton, Alain, 1998. "Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 38(3), pages 263-274, June.
    6. Andreas Neuenkirch & Samy Tindel, 2014. "A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise," Statistical Inference for Stochastic Processes, Springer, vol. 17(1), pages 99-120, April.
    7. Alexandra Chronopoulou & Samy Tindel, 2013. "On inference for fractional differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 29-61, April.
    8. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
    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. Rachid Belfadli & Khalifa Es-Sebaiy & Fatima-Ezzahra Farah, 2022. "Statistical analysis of the non-ergodic fractional Ornstein–Uhlenbeck process with periodic mean," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(7), pages 885-911, October.
    2. Nakajima, Shohei & Shimizu, Yasutaka, 2022. "Asymptotic normality of least squares type estimators to stochastic differential equations driven by fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 187(C).

    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. Xu, Weijun & Sun, Qi & Xiao, Weilin, 2012. "A new energy model to capture the behavior of energy price processes," Economic Modelling, Elsevier, vol. 29(5), pages 1585-1591.
    2. Zhang, Pu & Xiao, Wei-lin & Zhang, Xi-li & Niu, Pan-qiang, 2014. "Parameter identification for fractional Ornstein–Uhlenbeck processes based on discrete observation," Economic Modelling, Elsevier, vol. 36(C), pages 198-203.
    3. Marie, Nicolas, 2020. "Nonparametric estimation of the trend in reflected fractional SDE," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. Es-Sebaiy, Khalifa & Viens, Frederi G., 2019. "Optimal rates for parameter estimation of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3018-3054.
    5. Katsuto Tanaka, 2013. "Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 16(3), pages 173-192, October.
    6. Qian Yu, 2021. "Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean for general Hurst parameter," Statistical Papers, Springer, vol. 62(2), pages 795-815, April.
    7. Pavel Kříž & Leszek Szała, 2020. "Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes," Mathematics, MDPI, vol. 8(5), pages 1-20, May.
    8. Katsuto Tanaka, 2015. "Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 315-332, October.
    9. Bertin, Karine & Torres, Soledad & Tudor, Ciprian A., 2011. "Drift parameter estimation in fractional diffusions driven by perturbed random walks," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 243-249, February.
    10. Kohei Chiba, 2020. "An M-estimator for stochastic differential equations driven by fractional Brownian motion with small Hurst parameter," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 319-353, July.
    11. Nakajima, Shohei & Shimizu, Yasutaka, 2022. "Asymptotic normality of least squares type estimators to stochastic differential equations driven by fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 187(C).
    12. Katsuto Tanaka & Weilin Xiao & Jun Yu, 2020. "Maximum Likelihood Estimation for the Fractional Vasicek Model," Econometrics, MDPI, vol. 8(3), pages 1-28, August.
    13. Hui Jiang & Jingying Zhou, 2023. "An Exponential Nonuniform Berry–Esseen Bound for the Fractional Ornstein–Uhlenbeck Process," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1037-1058, June.
    14. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.
    15. Cai, Chunhao & Lv, Wujun, 2020. "Adaptative design for estimation of parameter of second order differential equation in fractional diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    16. Douissi, Soukaina & Es-Sebaiy, Khalifa & Alshahrani, Fatimah & Viens, Frederi G., 2022. "AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 886-918.
    17. Sun, Qi & Xu, Weijun & Xiao, Weilin, 2013. "An empirical estimation for mean-reverting coal prices with long memory," Economic Modelling, Elsevier, vol. 33(C), pages 174-181.
    18. Xichao Sun & Litan Yan & Yong Ge, 2022. "The Laws of Large Numbers Associated with the Linear Self-attracting Diffusion Driven by Fractional Brownian Motion and Applications," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1423-1478, September.
    19. Fabienne Comte & Nicolas Marie, 2021. "Nonparametric estimation for I.I.D. paths of fractional SDE," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 669-705, October.
    20. Herold Dehling & Brice Franke & Jeannette H. C. Woerner, 2017. "Estimating drift parameters in a fractional Ornstein Uhlenbeck process with periodic mean," Statistical Inference for Stochastic Processes, Springer, vol. 20(1), pages 1-14, April.

    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:129:y:2019:i:8:p:2880-2902. 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.