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

In: Modern Stochastics and Applications

Author

Listed:
  • Yuliya Mishura

    (Taras Shevchenko National University of Kyiv)

  • Kostiantyn Ral’chenko

    (Taras Shevchenko National University of Kyiv)

  • Oleg Seleznev

    (University of Umea)

  • Georgiy Shevchenko

    (Taras Shevchenko National University of Kyiv)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:spochp:978-3-319-03512-3_17
    DOI: 10.1007/978-3-319-03512-3_17
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    Cited by:

    1. 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.

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