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Multidimensional Levy walk and its scaling limits

Author

Listed:
  • Marek Teuerle
  • Piotr Zebrowski
  • Marcin Magdziarz

Abstract

In this paper we obtain the scaling limit of multidimensional Levy walk and describe the detailed structure of the limiting process. It occurs that the scaling limit is a subordinated alpha-stable Levy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method of simulating Levy walks with predefined spectral measure, which controls the direction of each jump. Our approach can be applied in the analysis of real-life data - we are able to recover the spectral measure from the data and obtain the full characterization of Levy walk. We also give examples of some useful spectral measures, which cover large class of possible scenarios in the modeling of real-life phenomena.

Suggested Citation

  • Marek Teuerle & Piotr Zebrowski & Marcin Magdziarz, 2011. "Multidimensional Levy walk and its scaling limits," HSC Research Reports HSC/11/06, Hugo Steinhaus Center, Wroclaw University of Science and Technology.
    • Handle: RePEc:wuu:wpaper:hsc1106
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    File URL: http://www.im.pwr.wroc.pl/~hugo/RePEc/wuu/wpaper/HSC_11_06.pdf
    File Function: Original version, 2011
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    References listed on IDEAS

    as
    1. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook9401, December.
    2. Joanna Janczura & Rafal Weron, 2011. "Black swans or dragon kings? A simple test for deviations from the power law," Papers 1102.3712, arXiv.org.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Magdziarz, M. & Scheffler, H.P. & Straka, P. & Zebrowski, P., 2015. "Limit theorems and governing equations for Lévy walks," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4021-4038.

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    More about this item

    Keywords

    Levy walk; scaling limit; convergence in distribution; spectral measure;
    All these keywords.

    JEL classification:

    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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