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Asymptotic Behaviour of the Distribution Density of the Fractional Lévy Motion

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Modern Stochastics and Applications

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

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Abstract

We investigate the distribution properties of the fractional Lévy motion defined by the Mandelbrot-Van Ness representation:

$$\displaystyle{Z_{t}^{H}:=\int _{ \mathbb{R}}f(t,s)dZ_{s},}$$

where Z s , \(s \in \mathbb{R}\), is a (two-sided) real-valued Lévy process, and

$$\displaystyle{f(t,s):= \frac{1} {\varGamma (H + 1/2)}\left [(t - s)_{+}^{H-1/2} - (-s)_{ +}^{H-1/2}\right ],\quad t,s \in \mathbb{R}.}$$

We consider separately the cases 0 < H < 1∕2 (short memory) and 1∕2 < H < 1 (long memory), where H is the Hurst parameter, and present the asymptotic behaviour of the distribution density of the process. Some examples are provided, in which it is shown that the behaviour of the density in the cases 0 < H < 1∕2 and 1∕2 < H < 1 is completely different.

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Acknowledgements

The authors thank the referee for helpful remarks and gratefully acknowledge the DFG Grant Schi 419/8-1. The first-named author also gratefully acknowledges the Scholarship of the President of Ukraine for young scientists (2011–2013).

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Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, Ukrainian National Academy of Sciences, Academy Glushkov Avenue 40, Kiev, 03187, Ukraine

    Victoria P. Knopova

  2. Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska street 3, Kiev, 01601, Ukraine

    Alexey M. Kulik

Authors
  1. Victoria P. Knopova
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  2. Alexey M. Kulik
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Correspondence to Victoria P. Knopova .

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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Knopova, V.P., Kulik, A.M. (2014). Asymptotic Behaviour of the Distribution Density of the Fractional Lévy 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_11

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