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
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
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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
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References
- 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
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980. MR 669666
- Y. Hu and D. Nualart, Parameter estimation for fractional Ornstein–Uhlenbeck processes, Stat. Probab. Lett. 8 (2010), 1030–1038. MR 2638974
- 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
- Y. Kozachenko, A. Melnikov and Y. Mishura, On drift parameter estimation in models with fractional Brownian motion, Statistics 49 (2015), no. 1. MR 3304366
- Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes Math., Springer, vol. 1929, 2008. MR 2378138
- D. Nualart and A. Rascanu, Differential equation driven by fractional Brownian motion, Collect. Math. 53 (2002), 55–81. MR 1893308
- S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, New York, 1993. MR 1347689
- C. A. Tudor and F. G. Viens, Statistical aspects of the fractional stochastic calculus, Ann. Stat. 35 (2007), 1183–1212. MR 2341703
- 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
- M. Zähle, Integration with respect to fractal functions and stochastic calculus, I. Prob. Theory Rel. Fields 111 (1998), 333–374. MR 1640795
- M. Zähle, On the link between fractional and stochastic calculus, Stochastic Dynamics, 1999, pp. 305–325. MR 1678495
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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
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
