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Parameter Estimation for Gaussian Processes with Application to the Model with Two Independent Fractional Brownian Motions

  • Yuliya Mishura
  • Kostiantyn Ralchenko
  • Sergiy Shklyar
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)

Abstract

The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form \(X_t = \theta G(t) + B_t\), where B is a Gaussian process, G(t) is a known function, and \(\theta \) is an unknown drift parameter. The estimation techniques for the cases of discrete-time and continuous-time observations are presented. As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered. Secondly, we study in detail the model with two independent fractional Brownian motions and apply the general results mentioned above to this model.

Keywords

Discrete observations Continuous observations Maximum likelihood estimator Strong consistency Fractional Brownian motion Fredholm integral equation of the first kind 

Notes

Acknowledgements

The research of Yu. Mishura was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758). Yu. Mishura and K. Ralchenko acknowledge that the present research is carried through within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Yuliya Mishura
    • 1
  • Kostiantyn Ralchenko
    • 1
  • Sergiy Shklyar
    • 1
  1. 1.Department of Probability Theory, Statistics and Actuarial MathematicsTaras Shevchenko National University of KyivKyivUkraine

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