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

  • Marco Dozzi
  • Yuriy Kozachenko
  • Yuliya Mishura
  • Kostiantyn RalchenkoEmail author


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.


Gaussian process Multifractional Brownian motion Parameter estimation Consistency Strong consistency Stochastic differential equation 

Mathematics Subject Classification

60G15 60G22 62F10 62F12 



The authors are grateful to the anonymous referees for their useful remarks and suggestions which contributed to a substantial improvement of the text.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institut Élie CartanUniversité de LorraineVandoeuvre-les-Nancy CedexFrance
  2. 2.Department of Probability, Statistics and Actuarial Mathematics, Mechanics and Mathematics FacultyTaras Shevchenko National University of KyivKyivUkraine

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