Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
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