Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 99 (2018).
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The fractional Vašíček model

$\displaystyle dX_t = \left (\alpha - \beta X_t \right )\,dt + \gamma \,dB_t^H$    

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 [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965. MR 415956
  • [2] R. Belfadli, K. Es-Sebaiy, and Y. Ouknine, Parameter estimation for fractional Ornstein-Uhlenbeck processes: non-ergodic case, Frontiers in Science and Engineering 1 (2011), 1-16.
  • [3] C. Berzin, A. Latour, and J. R. León, Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion, Springer International Publishing Switzerland, Cham, 2014. MR 3289986
  • [4] C. Berzin and J. R. León, Estimation in models driven by fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 191-213. MR 2446320
  • [5] P. Cheridito, H. Kawaguchi, and M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003). MR 1961165
  • [6] A. Chronopoulou and F. G. Viens, Estimation and pricing under long-memory stochastic volatility, Ann. Finance 8 (2012), no. 2-3, 379-403. MR 2922802
  • [7] A. Chronopoulou and F. G. Viens, Stochastic volatility and option pricing with long-memory in discrete and continuous time, Quantitative Finance 12 (2012), no. 4, 635-649. MR 2909603
  • [8] F. Comte, L. Coutin, and E. Renault, Affine fractional stochastic volatility models, Ann. Finance 8 (2012), no. 2-3, 337-378. MR 2922801
  • [9] F. Comte and E. Renault, Long memory in continuous-time stochastic volatility models, Math. Finance 8 (1998), no. 4, 291-323. MR 1645101
  • [10] S. Corlay, J. Lebovits, and J. L. Lévy Véhel, Multifractional stochastic volatility models, Math. Finance 24 (2014), no. 2, 364-402. MR 3274947
  • [11] C. C. Craig, On the frequency function of $ xy$, Ann. Math. Statist. 7 (1936), no. 1, 1-15.
  • [12] L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potent. Anal. 10 (1999), 177-214. MR 1677455
  • [13] M. El Machkouri, K. Es-Sebaiy, and Y. Ouknine, Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes, J. Korean Statist. Soc. 45 (2016), 329-341. MR 3527650
  • [14] H. Fink, C. Klüppelberg, and M. Zähle, Conditional distributions of processes related to fractional Brownian motion, J. Appl. Probab. 50 (2013), no. 1, 166-183. MR 3076779
  • [15] R. Hao, Y. Liu, and S. Wang, Pricing credit default swap under fractional Vasicek interest rate model, J. Math. Finance 4 (2014), no. 1, 10-20. MR 3597781
  • [16] Y. Hu and D. Nualart, Parameter estimation for fractional Ornstein-Uhlenbeck processes, Stat. Probab. Lett. 80 (2010), no. 11-12, 1030-1038. MR 2638974
  • [17] J. Istas and G. Lang, Quadratic variations and estimation of the local Hölder index of a Gaussian process, Ann. Inst. Henri Poincaré 33 (1997), no. 4, 407-436. MR 1465796
  • [18] M. Kleptsyna, A. Le Breton, and M.-C. Roubaud, Parameter estimation and optimal filtering for fractional type stochastic systems, Stat. Inference Stoch. Process. 3 (2000), 173-182. MR 1819294
  • [19] M. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process, Stat. Inference Stoch. Process. 5 (2002), no. 3, 229-248. MR 1943832
  • [20] 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
  • [21] 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
  • [22] K. Kubilius and Y. Mishura, The rate of convergence of Hurst index estimate for the stochastic differential equation, Stoch. Process. Appl. 122 (2012), no. 11, 3718-3739. MR 2965922
  • [23] K. Kubilius, Y. Mishura, and K. Ralchenko, Parameter Estimation in Fractional Diffusion Models, Springer International Publishing AG, Cham, 2017. MR 3752152
  • [24] K. Kubilius, V. Skorniakov, and 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. MR 3608077
  • [25] Y. A. Kutoyants, Statistical inference for ergodic diffusion processes, Springer, London, 2004. MR 2144185
  • [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. MR 1629915
  • [27] R. Liptser and A. Shiryayev, Theory of Martingales, Kluwer Academic Publishers, Dordrecht, 1989. MR 1022664
  • [28] S. Lohvinenko, K. Ralchenko, and O. Zhuchenko, Asymptotic properties of parameter estimators in fractional Vašíček model, Lithuanian J. Statist. 55 (2016), no. 1, 102-111.
  • [29] S. Lohvinenko and K. Ralchenko, Maximum likelihood estimation in the fractional Vašíček model, Lithuanian J. Statist. 56 (2017), no. 1, 77-87.
  • [30] 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
  • [31] I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999), no. 4, 571-587. MR 1704556
  • [32] K. B. Oldham, J. C. Myland, and J. Spanier, An atlas of functions: with Equator, the atlas function calculator, Springer-Verlag, New York, 2009. MR 2466333
  • [33] B. Prakasa Rao, Statistical Inference for Fractional Diffusion Processes, Wiley, Chichester, 2010. MR 2778592
  • [34] L. Song and K. Li, Pricing option with stochastic interest rates and transaction costs in fractional Brownian markets, Discrete Dynamics in Nature and Society 2018 (2018). MR 3842704
  • [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. MR 3123562
  • [36] 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
  • [37] O. Vašíček, 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. MR 1349110
  • [39] 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_research/1861.
  • [40] 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_research/1966. MR 3904176
  • [41] 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_research/2158 MR 3904332
  • [42] 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
  • [43] 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

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60G22, 62F10, 62F12

Retrieve articles in all journals with MSC (2010): 60G22, 62F10, 62F12


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

DOI: https://doi.org/10.1090/tpms/1087
Keywords: Fractional Brownian motion, fractional Va\v{s}\'{\i}\v{c}ek 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