Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

Asymptotic properties of non-standard drift parameter estimators in the models involving fractional Brownian motion


Authors: Meriem Bel Hadj Khlifa, Yuliya Mishura and Mounir Zili
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
Journal: Theor. Probability and Math. Statist. 94 (2017), 77-88
MSC (2010): Primary 62F10, 62F12; Secondary 60G22
DOI: https://doi.org/10.1090/tpms/1010
Published electronically: August 25, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the problem of estimation of the unknown drift parameter in the stochastic differential equations driven by fractional Brownian motion, with the coefficients supplying standard existence-uniqueness demands. We consider a particular case when the ratio of drift and diffusion coefficients is non-random, and establish the asymptotic strong consistency of the estimator with different ratios, from many classes of non-random standard functions. Simulations are provided to illustrate our results, and they demonstrate the fast rate of convergence of the estimator to the true value of a parameter.


References [Enhancements On Off] (What's this?)

  • [1] K. Bertin, S. Torres, and C. Tudor, Drift parameter estimation in fractional diffusions driven by perturbed random walks, Stat. Probab. Lett. 81 (2011), 243-249. MR 2764290
  • [2] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980. MR 669666
  • [3] Y. Hu and D. Nualart, Parameter estimation for fractional Ornstein-Uhlenbeck processes, Stat. Probab. Lett. 8 (2010), 1030-1038. MR 2638974
  • [4] M. L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process, Stat. Inference Stoch. Process. 5 (2002), 229-248. MR 1943832
  • [5] Y. Kozachenko, A. Melnikov and Y. Mishura, On drift parameter estimation in models with fractional Brownian motion, Statistics 49 (2015), no. 1. MR 3304366
  • [6] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes Math., Springer, vol. 1929, 2008. MR 2378138
  • [7] D. Nualart and A. Rascanu, Differential equation driven by fractional Brownian motion, Collect. Math. 53 (2002), 55-81. MR 1893308
  • [8] S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, New York, 1993. MR 1347689
  • [9] C. A. Tudor and F. G. Viens, Statistical aspects of the fractional stochastic calculus, Ann. Stat. 35 (2007), 1183-1212. MR 2341703
  • [10] W. Xiao, W. Zhang, and W. Xu, Parameter estimation for fractional OrnsteinUhlenbeck processes at discrete observation, Appl. Math. Modell. 35 (2011), 4196-4207. MR 2801946
  • [11] M. Zähle, Integration with respect to fractal functions and stochastic calculus, I. Prob. Theory Rel. Fields 111 (1998), 333-374. MR 1640795
  • [12] M. Zähle, On the link between fractional and stochastic calculus, Stochastic Dynamics, 1999, pp. 305-325. MR 1678495

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 62F10, 62F12, 60G22

Retrieve articles in all journals with MSC (2010): 62F10, 62F12, 60G22


Additional Information

Meriem Bel Hadj Khlifa
Affiliation: Faculty of Sciences of Monastir, Department of Mathematics, Avenue de l’Environnement, 5000, Monastir, Tunisia
Email: meriem.bhk17121988@outlook.fr

Yuliya Mishura
Affiliation: Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska Street, 64, 01601 Kyiv, Ukraine
Email: myus@univ.kiev.ua

Mounir Zili
Affiliation: Faculty of Sciences of Monastir, Department of Mathematics, Avenue de l’Environnement, 5000, Monastir, Tunisia
Email: Mounir.Zili@fsm.rnu.tn

DOI: https://doi.org/10.1090/tpms/1010
Keywords: Parameter estimators, fractional Brownian motion, strong consistency, estimation of fractional derivatives
Received by editor(s): March 17, 2016
Published electronically: August 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society