IDEAS home Printed from https://ideas.repec.org/a/spr/sistpr/v21y2018i3d10.1007_s11203-017-9161-9.html
   My bibliography  Save this article

Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion

Author

Listed:
  • Antoine Lejay

    (Université de Lorraine, IECL, UMR 7502
    CNRS, IECL, UMR 7502
    Inria)

Abstract

We study the asymptotic property of simple estimator of the parameter of a skew Brownian motion when one observes its positions on a fixed grid—or equivalently of a simple random walk with a bias at 0. This estimator, nothing more than the maximum likelihood estimator, is based only on the number of passages of the random walk at 0. It is very simple to set up, is consistent and is asymptotically mixed normal. We believe that this simplified framework is helpful to understand the asymptotic behavior of the maximum likelihood of the skew Brownian motion observed at discrete times which is studied in a companion paper.

Suggested Citation

  • Antoine Lejay, 2018. "Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 539-551, October.
  • Handle: RePEc:spr:sistpr:v:21:y:2018:i:3:d:10.1007_s11203-017-9161-9
    DOI: 10.1007/s11203-017-9161-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11203-017-9161-9
    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/s11203-017-9161-9?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. Fernholz, E. Robert & Ichiba, Tomoyuki & Karatzas, Ioannis, 2013. "Two Brownian particles with rank-based characteristics and skew-elastic collisions," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2999-3026.
    2. Danielle Florens, 1998. "Estimation of the Diffusion Coefficient from Crossings," Statistical Inference for Stochastic Processes, Springer, vol. 1(2), pages 175-195, May.
    3. Luis H. R. Alvarez E. & Paavo Salminen, 2017. "Timing in the presence of directional predictability: optimal stopping of skew Brownian motion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 377-400, October.
    4. repec:dau:papers:123456789/1908 is not listed on IDEAS
    5. Bass, Richard F. & Khoshnevisan, Davar, 1993. "Rates of convergence to Brownian local time," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 197-213, September.
    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. Andrey Sarantsev, 2014. "On a class of diverse market models," Annals of Finance, Springer, vol. 10(2), pages 291-314, May.
    2. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    3. Antoine Lejay & Paolo Pigato, 2017. "Data and methods for A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data [Données et méthodes pour "A threshold model for local volatilit," Working Papers hal-01668975, HAL.
    4. Haoyan Zhang & Yingxu Tian, 2022. "Hitting Time Problems of Sticky Brownian Motion and Their Applications in Optimal Stopping and Bond Pricing," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1237-1251, June.
    5. Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Post-Print hal-00921151, HAL.
    6. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 2011. "On the local time of random walk on the 2-dimensional comb," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1290-1314, June.
    7. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    8. Benjamin Jourdain & Julien Reygner, 2013. "Capital distribution and portfolio performance in the mean-field Atlas model," Papers 1312.5660, arXiv.org, revised Aug 2014.
    9. Robert Fernholz, 2017. "Stratonovich representation of semimartingale rank processes," Papers 1705.00336, arXiv.org.
    10. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    11. Atar, Rami & Budhiraja, Amarjit, 2015. "On the multi-dimensional skew Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1911-1925.
    12. Peter Grandits & Maike Klein, 2020. "Ruin probability in a two-dimensional model with correlated Brownian motions," Papers 2004.13601, arXiv.org.
    13. Lempa, Jukka & Mordecki, Ernesto & Salminen, Paavo, 2024. "Diffusion spiders: Green kernel, excessive functions and optimal stopping," Stochastic Processes and their Applications, Elsevier, vol. 167(C).

    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:sistpr:v:21:y:2018:i:3:d:10.1007_s11203-017-9161-9. 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.