Asymptotic distribution of the maximum likelihood estimator in the fractional Vašíček model
Authors:
S. S. Lohvinenko and K. V. Ralchenko
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 99 (2019), 149-168
MSC (2010):
Primary 60G22, 62F10, 62F12
DOI:
https://doi.org/10.1090/tpms/1087
Published electronically:
February 27, 2020
MathSciNet review:
3908663
Full-text PDF
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References |
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Additional Information
Abstract: The fractional Vašíček model \begin{equation*} dX_t = \left (\alpha - \beta X_t \right ) dt + \gamma dB_t^H \end{equation*} is considered. The model is driven by the fractional Brownian motion $B^H$ with the Hurst index $H\in \bigl (\frac 12,1\bigr )$. The asymptotic distribution of the maximum likelihood estimator is studied for the vector parameter $(\alpha , \beta )$. It is proved that this estimator is asymptotically normal in the case of $\beta >0$. It is shown that the estimators of the parameters $\alpha$ and $\beta$ are asymptotically independent.
References
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- Y. Kozachenko, A. Melnikov, and Y. Mishura, On drift parameter estimation in models with fractional Brownian motion, Statistics 49 (2015), no. 1, 35–62. MR 3304366
- K. Kubilius and 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. MR 2738897
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- 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. MR 1629915
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- S. Lohvinenko and K. Ralchenko, Maximum likelihood estimation in the fractional Vašíček model, Lithuanian J. Statist. 56 (2017), no. 1, 77–87.
- Y. Mishura and K. Ralchenko, Drift parameter estimation in the models involving fractional Brownian motion, International Conference on Modern Problems of Stochastic Analysis and Statistics, 2016, pp. 237–268. MR 3747669
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- 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. MR 3123562
- K. Tanaka, Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process, Stat. Inf. Stoch. Process. 18 (2015), no. 3, 315–332. MR 3395610
- O. Vašíček, An equilibrium characterization of the term structure, J. Finance Econ. 5 (1977), no. 2, 177–188.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995. MR 1349110
- W. Xiao and J. Yu, Asymptotic theory for estimating the persistent parameter in the fractional Vašíček model, CSR for Sustainability and Success: Corporate Social Responsibility in Singapore. Research Collection School Of Economics (2016), 1–27; http://ink.library.smu.edu.sg/soe_{r}esearch/1861.
- W. Xiao and J. Yu, Asymptotic theory for estimating drift parameters in the fractional Vašíček model, Research Collection School Of Economics (2017); https://ink.library.smu.edu.sg/soe_{r}esearch/1966. MR 3904176
- W. Xiao and J. Yu, Asymptotic theory for rough fractional Vašíček models, Econom. Letters, 2019 April, Volume 177, Pages 26–29; https://ink.library.smu.edu.sg/soe_{r}esearch/2158 MR 3904332
- W. Xiao, W. Zhang, X. Zhang, and X. Chen, The valuation of equity warrants under the fractional Vasicek process of the short-term interest rate, Phys. A 394 (2014), 320–337. MR 3123524
- F. Yerlikaya-Özkurt, C. Vardar-Acar, Y. Yolcu-Okur, and G. W. Weber, Estimation of the Hurst parameter for fractional Brownian motion using the CMARS method, J. Comput. Appl. Math. 259 (2014), 843–850. MR 3132849
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Additional Information
S. S. Lohvinenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
stanislav.lohvinenko@gmail.com
K. V. Ralchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
k.ralchenko@gmail.com
Keywords:
Fractional Brownian motion,
fractional Vašíček model,
maximum likelihood estimators,
moment generating function,
asymptotic distribution
Received by editor(s):
October 8, 2018
Published electronically:
February 27, 2020
Additional Notes:
The research of the second author was done in the framework and under the support of the project “STORM: Stochastics for Time-Space Risk Models”, a Toppforsk project funded by the Norwegian Research Council in cooperation with the University of Oslo.
Article copyright:
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