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Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation

Author

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  • Marco Dozzi

    (Université de Lorraine)

  • Yuriy Kozachenko

    (Taras Shevchenko National University of Kyiv)

  • Yuliya Mishura

    (Taras Shevchenko National University of Kyiv)

  • Kostiantyn Ralchenko

    (Taras Shevchenko National University of Kyiv)

Abstract

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.

Suggested Citation

  • Marco Dozzi & Yuriy Kozachenko & Yuliya Mishura & Kostiantyn Ralchenko, 2018. "Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 21-52, April.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:1:d:10.1007_s11203-016-9147-z
    DOI: 10.1007/s11203-016-9147-z
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    References listed on IDEAS

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    1. Yuliya Mishura & Kostiantyn Ral’chenko & Oleg Seleznev & Georgiy Shevchenko, 2014. "Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 303-318, Springer.
    2. Stoev, Stilian A. & Taqqu, Murad S., 2006. "How rich is the class of multifractional Brownian motions?," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 200-221, February.
    3. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
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