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

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Stochastic Processes and Applications (SPAS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

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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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Authors and Affiliations

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

    Yuliya Mishura, Kostiantyn Ralchenko & Sergiy Shklyar

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  1. Yuliya Mishura
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  2. Kostiantyn Ralchenko
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  3. Sergiy Shklyar
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Correspondence to Yuliya Mishura .

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Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Mishura, Y., Ralchenko, K., Shklyar, S. (2018). Parameter Estimation for Gaussian Processes with Application to the Model with Two Independent Fractional Brownian Motions. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Stochastic Processes and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-030-02825-1_6

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