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

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. Danielle Florens, 1998. "Estimation of the Diffusion Coefficient from Crossings," Statistical Inference for Stochastic Processes, Springer, vol. 1(2), pages 175-195, May.
    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.
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