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

Tomasz R. Werner 1Ryszard Kutner 1Tomasz Gubiec 1Didier Sornette 2

1. Faculty of Physics, University of Warsaw (FPUW), Pasteura 5, Warsaw 02-093, Poland
2. ETH Zurich and Swiss Finance Institute, Kreuzplatz 5, Zurich CH-8032, Switzerland


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.

The European Physical Journal Special Topics 205 (2012) 27-52.


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Submitted: 2016-03-29 12:43
Revised:   2016-03-29 12:48