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

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Modern Stochastics and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 90))

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.

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Author information

Authors and Affiliations

  1. Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601, Kyiv, Ukraine

    Yuliya Mishura, Kostiantyn Ral’chenko & Georgiy Shevchenko

  2. Institute of Mathematics and Mathematical Statistics, University of Umea, SE-90187, Umea, Sweden

    Oleg Seleznev

Authors
  1. Yuliya Mishura
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  2. Kostiantyn Ral’chenko
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  3. Oleg Seleznev
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  4. Georgiy Shevchenko
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Corresponding author

Correspondence to Georgiy Shevchenko .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Volodymyr Korolyuk

  2. Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, Compiegne Cedex, France

    Nikolaos Limnios

  3. Dept. of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Yuliya Mishura

  4. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Lyudmyla Sakhno

  5. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Georgiy Shevchenko

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Mishura, Y., Ral’chenko, K., Seleznev, O., Shevchenko, G. (2014). Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_17

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