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Theory of Probability and Mathematical Statistics

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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
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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.


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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: © Copyright 2020 American Mathematical Society