Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion
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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.
KeywordsSkew random walk Skew Brownian motion Maximum likelihood estimator Local asymptotic mixed normality Local time Null recurrent process
This work has been developed within the framework of the Inria’s Équipe Associée ANESTOC-TOSCA between France and Chile. It is associated to a joint work with E. Mordecki and S. Torres on the estimation of the parameter of the Skew Brownian motion. The author wishes to thank them for interesting discussion on this topic. The author also wishes the referees for their careful reading and having suggested corrections, improvements and for having pointing out the attention toward Bayesian estimation.
- Alvarez LHR, Salminen P (2016) Timing in the presence of directional predictability: optimal stopping of Skew Brownian Motion (2016). Preprint arxiv:1608.04537
- Barahona M, Rifo L, Sepúlveda M, Torres S (2016) A simulation-based study on Bayesian estimators for the skew Brownian motion. Entropy 18.7, Paper No. 241, 14. issn: 1099-4300. doi: 10.3390/e18070241
- Ibragimov IA, Has’ minskiĭ RZ (1981) Statistical estimation. Asymptotic theory. Applications of mathematics. Springer, New YorkGoogle Scholar
- Le Gall J-F (1985) One-dimensional stochastic differential equations involving the local times of the unknown process. In: Stochastic analysis and applications. Vol. 1095. Lecture Notes in Mathematics. Springer Verlag, 51–82Google Scholar
- Lejay A Pigato P (2017) Statistical estimation of the Oscillating Brownian Motion. Preprint arxiv:1701.02129
- Lejay A, Mordecki E, Torres S (2017) Two consistent estimators for the Skew Brownian motion. PreprintGoogle Scholar
- Lipton A, Sepp A (2011) Filling the Gap. Risk (Oct. 2011), 86–91Google Scholar
- Revuz D, Yor M (1999) Continuous martingales and Brownian motion. 3rd ed. Vol. 293. Grundlehren der Mathematischen Wissenschaften.Springer-Verlag, Berlin, doi: 10.1007/978-3-662-06400-9
- Robert CP (2007) The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer texts in statistics, 2nd edn. Springer, New YorkGoogle Scholar
- Walsh J (1978) A diffusion with discontinuous local time. In: Temps locaux. Vol. 52–53. Astérisques. SociétéMathématique de France, 37–45Google Scholar