Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion

  • Yuliya Mishura
  • Kostiantyn Ral’chenko
  • Oleg Seleznev
  • Georgiy ShevchenkoEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 90)


In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.


Stochastic Differential Equation Fractional Derivative Maximum Likelihood Estimator Wiener Process Fractional Brownian Motion 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yuliya Mishura
    • 1
  • Kostiantyn Ral’chenko
    • 1
  • Oleg Seleznev
    • 2
  • Georgiy Shevchenko
    • 1
    Email author
  1. 1.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine
  2. 2.Institute of Mathematics and Mathematical StatisticsUniversity of UmeaUmeaSweden

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