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   Editors Board   Contacts   Template   Publication Ethics   Peer Review Process

Theory of Probability and Mathematical Statistics
(Teoriya imovirnostey ta matematychna statystyka)



Asymptotic distribution of maximum likelihood estimator in fractional Vasicek model

S. S. Lohvinenko, K. V. Ralchenko

Download PDF

Abstract: We consider the fractional Vasicek model of the form dXt=(α−βXt)dt+γdBtH driven by fractional Brownian motion BH with Hurst index H∈(1/2,1). We study asymptotic distribution of maximum likelihood estimator of vector parameter (α,β) and prove its asymptotic normality in the case β>0. We show that the estimators of α and β are asymptotically independent.

Keywords: Fractional Brownian motion, fractional Vasicek model, maximum likelihood estimator, moment-generating function, asymptotic distribution.

Bibliography:
1. M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, Dover publications, New York, 1965.
2. R. Belfadli, K. Es-Sebaiy, Y. Ouknine, Parameter estimation for fractional Ornstein–Uhlenbeck processes: non-ergodic case , Frontiers in Science and Engineering (An International Journal Edited by Hassan II Academy of Science and Technology), 1 (2011), 1-16.
3. C. Berzin, A. Latour, J.R. Leon, Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion , Springer, 2014.
4. C. Berzin, J.R. Leon, Estimation in models driven by fractional Brownian motion , Ann. Inst. Henri Poincare Probab. Stat., 44 (2008), no. 2, 191-213.
5. P. Cheridito, H. Kawaguchi, M. Maejima, Fractional Ornstein–Uhlenbeck processes , Electron. J. Probab., 8 (2003).
6. A. Chronopoulou, F. G. Viens, Estimation and pricing under long-memory stochastic volatility, Annals of Finance, 8 (2012), no. 2-3, 379-403.
7. A. Chronopoulou, F. G. Viens, Stochastic volatility and option pricing with long-memory in discrete and continuous time , Quantitative Finance, 12 (2012), no. 4, 635-649.
8. F. Comte, L. Coutin, E. Renault, Affine fractional stochastic volatility models , Annals of Finance, 8 (2012), no. 2-3, 337-378.
9. F. Comte, E. Renault, Long memory in continuous-time stochastic volatility models , Mathematical Finance, 8 (1998), no. 4, 291-323.
10. S. Corlay, J. Lebovits, J. L. Levy Vehel, Multifractional stochastic volatility models , Mathematical Finance, 24 (2014), no. 2, 364-402.
11. C. C. Craig, On the frequency function of xy , The Annals of Mathematical Statistics, 7 (1936), no. 1, 1-15.
12. L. Decreusefond, A. S. Ustunel, Stochastic analysis of the fractional Brownian motion, Potent. Anal., 10 (1999), 177-214.
13. M. El Machkouri, K. Es-Sebaiy, Y. Ouknine, Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes, Journal of the Korean Statistical Society, 45 (2016), 329-341.
14. H. Fink, C. Kluppelberg, M. Zahle, Conditional distributions of processes related to fractional Brownian motion , J. Appl. Probab., 50 (2013), no. 1, 166-183.
15. R. Hao, Y. Liu, S. Wang, Pricing credit default swap under fractional Vasicek interest rate model , Journal of Mathematical Finance, 4 (2014), no. 1, 10-20.
16. Y. Hu, D. Nualart, Parameter estimation for fractional Ornstein–Uhlenbeck processes , Stat. Probab. Lett., 80 (2010), no. 11-12, 1030-1038.
17. J. Istas, G. Lang, Quadratic variations and estimation of the local Holder index of a Gaussian process , Ann. Inst. Henri Poincare, 33 (1997), no. 4, 407-436.
18. M. Kleptsyna, A. Le Breton, M.-C. Roubaud, Parameter estimation and optimal filtering for fractional type stochastic systems, Stat. Inference Stoch. Process., 3 (2000), 173-182.
19. M. Kleptsyna, A. Le Breton, Statistical analysis of the fractional Ornstein–Uhlenbeck type process, Stat. Inference Stoch. Process., 5 (2002), no. 3, 229-248.
20. Y. Kozachenko, A. Melnikov, Y. Mishura, On drift parameter estimation in models with fractional Brownian motion, Statistics, 49 (2015), no. 1, 35-62.
21. K. Kubilius, D. Melichov, Quadratic variations and estimation of the Hurst index of the solution of SDE driven by a fractional Brownian motion, Lith. Math. J., 50 (2010), no. 4, 401-417.
22. K. Kubilius, Y. Mishura, The rate of convergence of Hurst index estimate for the stochastic differential equation, Stochastic Process. Appl., 122 (2012), no. 11, 3718-3739.
23. K. Kubilius, Y. Mishura, K. Ralchenko, Parameter Estimation in Fractional Diffusion Models, Springer, 2017.
24. K. Kubilius, V. Skorniakov, K. Ralchenko, The rate of convergence of the Hurst index estimate for a stochastic differential equation, Nonlinear Anal. Model. Control, 22 (2017), no. 2, 273-284.
25. Y. A. Kutoyants, Statistical inference for ergodic diffusion processes , Springer, London, 2004.
26. A. Le Breton, Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion , Statist. Probab. Lett., 38 (1998), no. 3, 263-274.
27. R. Liptser, A. Shiryayev, Theory of Martingales , Kluwer Academic Publishers, Dordrecht etc., 1989.
28. S. Lohvinenko, K. Ralchenko, O. Zhuchenko, Asymptotic properties of parameter estimators in fractional Vasicek model, Lithuanian J. Statist., 55 (2016), no. 1, 102-111.
29. S. Lohvinenko, K. Ralchenko, Maximum likelihood estimation in the fractional Vasicek model, Lithuanian J. Statist., 56 (2017), no. 1, 77-87.
30. Y. Mishura, K. Ralchenko, Drift Parameter Estimation in the Models Involving Fractional Brownian Motion, In International Conference on Modern Problems of Stochastic Analysis and Statistics, (2016), 237-268.
31. I. Norros, E. Valkeila, J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli, 5 (1999), no. 4, 571-587.
32. K. B. Oldham, J. C. Myland, J. Spanier, An atlas of functions: with Equator, the atlas function calculator , Springer-Verlag, New York, 2009.
33. B. Prakasa Rao, Statistical Inference for Fractional Diusion Processes, Wiley, Chichester, 2010.
34. L. Song, K. Li, Pricing Option with Stochastic Interest Rates and Transaction Costs in Fractional Brownian Markets , Discrete Dynamics in Nature and Society, 2018 (2018).
35. K. Tanaka, Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process , Stat. Inference Stoch. Process., 16 (2013), no. 3, 173-192.
36. K. Tanaka, Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process , Statistical Inference for Stochastic Processes, 18 (2015), no. 3, 315-332.
37. O. Vasicek, An equilibrium characterization of the term structure , J. Finance Econ., 5 (1977), no. 2, 177-188.
38. G. N. Watson, A Treatise on the Theory of Bessel Functions , Cambridge University Press, Cambridge, 1995.
39. W. Xiao, J. Yu, Asymptotic theory for estimating the persistent parameter in the fractional Vasicek model , CSR for Sustainability and Success: Corporate Social Responsibility in Singapore. Research Collection School Of Economics, (2016), 1-27. Available at: http://ink.library.smu.edu.sg/soe_research/1861
40. W. Xiao, J. Yu, Asymptotic theory for estimating drift parameters in the fractional Vasicek model , Research Collection School Of Economics, (2017). Available at: https://ink.library.smu.edu.sg/soe_research/1966
41. W. Xiao, J. Yu, Asymptotic theory for rough fractional vasicek models , Research Collection School Of Economics, (2018), 1-15. Available at: https://ink.library.smu.edu.sg/soe_research/2158
42. W. Xiao, W. Zhang, X. Zhang, X. Chen, The valuation of equity warrants under the fractional Vasicek process of the short-term interest rate, Phys. A, 394 (2014), 320-337.
43. F. Yerlikaya-Ozkurt, C. Vardar-Acar, Y. Yolcu-Okur, G.W. Weber, Estimation of the Hurst parameter for fractional Brownian motion using the CMARS method , Journal of Computational and Applied Mathematics, 259 (2014), 843-850.