IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v21y2019i4d10.1007_s11009-018-9654-z.html
   My bibliography  Save this article

The Least Squares Estimation for the α-Stable Ornstein-Uhlenbeck Process with Constant Drift

Author

Listed:
  • Yurong Pan

    (Bengbu University)

  • Litan Yan

    (Donghua University)

Abstract

In this paper, we consider the least squares estimators of the Ornstein-Uhlenbeck process with a constant drift dXt=(θ1−θ2Xt)dt+dZt$$dX_{t}=(\theta_{1}-\theta_{2}X_{t})dt+dZ_{t} $$with X0 = x0, where θ1, θ2 are two unknown parameters with θ2 > 0 and Z is a strictly symmetric α-stable motion on ℝ with the index α ∈ (1, 2). We construct the least squares estimators of θ1 and θ2 based on the discrete observation, and discuss the strong consistency and asymptotic distributions of the two estimators. Finally, we give some numerical calculus and simulations.

Suggested Citation

  • Yurong Pan & Litan Yan, 2019. "The Least Squares Estimation for the α-Stable Ornstein-Uhlenbeck Process with Constant Drift," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1165-1182, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9654-z
    DOI: 10.1007/s11009-018-9654-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-018-9654-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-018-9654-z?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. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    2. Shibin Zhang & Xinsheng Zhang, 2013. "A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 89-103, February.
    3. Li, Zenghu & Ma, Chunhua, 2015. "Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3196-3233.
    Full references (including those not matched with items on IDEAS)

    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. Yiying Cheng & Yaozhong Hu & Hongwei Long, 2020. "Generalized moment estimators for $$\alpha $$α-stable Ornstein–Uhlenbeck motions from discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 53-81, April.
    2. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    3. 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.
    4. Ren, Panpan & Wu, Jiang-Lun, 2021. "Least squares estimation for path-distribution dependent stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    5. Mátyás Barczy & Kristóf Körmendi & Gyula Pap, 2016. "Statistical inference for critical continuous state and continuous time branching processes with immigration," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(7), pages 789-816, October.
    6. Zhang, Xuekang & Yi, Haoran & Shu, Huisheng, 2019. "Nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 8-16.
    7. Guangjun Shen & Qian Yu, 2019. "Least squares estimator for Ornstein–Uhlenbeck processes driven by fractional Lévy processes from discrete observations," Statistical Papers, Springer, vol. 60(6), pages 2253-2271, December.
    8. Hui He & Zenghu Li & Wei Xu, 2018. "Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1952-1974, December.
    9. Shen, Leyi & Xia, Xiaoyu & Yan, Litan, 2022. "Least squares estimation for the linear self-repelling diffusion driven by α-stable motions," Statistics & Probability Letters, Elsevier, vol. 181(C).
    10. Yanfeng Wu & Jianqiang Hu & Xiangyu Yang, 2022. "Moment estimators for parameters of Lévy‐driven Ornstein–Uhlenbeck processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(4), pages 610-639, July.
    11. Mathias Mørck Ljungdahl & Mark Podolskij, 2022. "Multidimensional parameter estimation of heavy‐tailed moving averages," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 593-624, June.
    12. Mitsuki Kobayashi & Yasutaka Shimizu, 2023. "Threshold estimation for jump-diffusions under small noise asymptotics," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 361-411, July.
    13. Yang, Xu, 2017. "Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 18-27.
    14. Ying Jiao & Chunhua Ma & Simone Scotti, 2017. "Alpha-CIR model with branching processes in sovereign interest rate modeling," Finance and Stochastics, Springer, vol. 21(3), pages 789-813, July.
    15. Aur'elien Alfonsi & Guillaume Szulda, 2024. "On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients," Papers 2402.19203, arXiv.org.
    16. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    17. Ying Jiao & Chunhua Ma & Simone Scotti & Chao Zhou, 2018. "The Alpha-Heston Stochastic Volatility Model," Papers 1812.01914, arXiv.org.
    18. 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).
    19. Fontana, Claudio & Gnoatto, Alessandro & Szulda, Guillaume, 2023. "CBI-time-changed Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 323-349.
    20. Ying Jiao & Chunhua Ma & Simone Scotti & Chao Zhou, 2021. "The Alpha‐Heston stochastic volatility model," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 943-978, July.

    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:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9654-z. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.