A Journal "Theory of Probability and Mathematical Statistics"
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editorial Board   Contacts   Template   Publication Ethics   Peer Review Process   Special Issues   History  

Theory of Probability and Mathematical Statistics



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

Meriem Bel Hadj Khlifa, Yuliya Mishura, and Mounir Zili

Download PDF

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.

Keywords: . Parameter estimators, fractional Brownian motion, strong consistency, estimation of fractional derivatives.

Bibliography:
1. K. Bertin, S. Torres, and C. Tudor, Drift parameter estimation in fractional diffusions driven by perturbed random walks, Statistics & Probability Letters 81 (2011), 243–249.
2. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
3. Y. Hu and D. Nualart, Parameter estimation for fractional Ornstein–Uhlenbeck processes, Statistics & Probability Letters 8 (2010), 1030–1038
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.
5. Y. Kozachenko, A. Melnikov and Y. Mishura, On drift parameter estimation in models with fractional Brownian motion, Statistics: A Journal of Theoretical and Applied Statistics 49 (2015), no. 1.
6. Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes Math., Springer, vol. 1929, 2008.
7. D. Nualart and A. Rascanu, Differential equation driven by fractional Brownian motion, Collect. Math. 53 (2002), 55–81.
8. S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, New York, 1993.
9. C. A. Tudor and F. G. Viens, Statistical aspects of the fractional stochastic calculus, Ann. Stat.35 (2007), 1183–1212.
10. W. Xiao, W. Zhang, and W. Xu, Parameter estimation for fractional OrnsteinUhlenbeck processes at discrete observation, Applied Mathematical Modelling 35 (2011), 4196–4207.
11. M. Z?ahle, Integration with respect to fractal functions and stochastic calculus, I. Prob. Theory Rel. Fields 111 (1998), 333–374.
12. M. Z?ahle, On the link between fractional and stochastic calculus, Stochastic Dynamics, 1999,pp. 305–325