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Modeling of super-extreme events: An application to the hierarchical Weierstrass-Mandelbrot Continuous-time Random Walk

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Abstract

We analytically demonstrate and numerically simulate two utmost cases of dragon-kings’ impact on the (unnormalized) velocity autocorrelation function (VACF) of a complex time series generated by stochastic random walker. The first type of dragon-kings corresponds to a sustained drift whose duration time is much longer than that of any other event. The second type of dragon-kings takes the form of an abrupt shock whose amplitude velocity is much larger than those corresponding to any other event. The stochastic process in which the dragon-kings occur corresponds to an enhanced diffusion generated within the hierarchical Weierstrass-Mandelbrot Continuous-time Random Walk (WM-CTRW) formalism. Our analytical formulae enable a detailed study of the impact of the two super-extreme events on the VACF calculated for a given random walk realization on the form of upward deviations from the background power law decay present in the absence of dragon-kings. This allows us to provide a unambiguous distinction between the super-extreme dragon-kings and ‘normal’ extreme “black swans”. The results illustrate diagnostic that could be useful for the analysis of extreme and super-extreme events in real empirical time series.

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Authors and Affiliations

  1. Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Hoża 69, 00681, Warsaw, Poland

    T. R. Werner

  2. Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Hoża 69, 00681, Warsaw, Poland

    T. Gubiec & R. Kutner

  3. Department of Management, Technology and Economics, ETH-Zürich, Kreuzplaz 5, 8032, Zürich, Switzerland

    D. Sornette

Authors
  1. T. R. Werner
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  2. T. Gubiec
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  3. R. Kutner
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  4. D. Sornette
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Werner, T.R., Gubiec, T., Kutner, R. et al. Modeling of super-extreme events: An application to the hierarchical Weierstrass-Mandelbrot Continuous-time Random Walk. Eur. Phys. J. Spec. Top. 205, 27–52 (2012). https://doi.org/10.1140/epjst/e2012-01560-0

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  • DOI: https://doi.org/10.1140/epjst/e2012-01560-0

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