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Modeling of superextreme events: An application to the hierarchical WeierstrassMandelbrot Continuoustime Random Walk 
Tomasz R. Werner ^{1}, Ryszard Kutner ^{1}, Tomasz Gubiec ^{1}, Didier Sornette ^{2} 
1. Faculty of Physics, University of Warsaw (FPUW), Pasteura 5, Warsaw 02093, Poland 
Abstract 
We analytically demonstrate and numerically simulate two utmost cases of dragonkings’ impact on the (unnormalized) velocity autocorrelation function (VACF) of a complex time series generated by stochastic random walker. The first type of dragonkings corresponds to a sustained drift whose duration time is much longer than that of any other event. The second type of dragonkings 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 dragonkings occur corresponds to an enhanced diffusion generated within the hierarchical WeierstrassMandelbrot Continuoustime Random Walk (WMCTRW) formalism. Our analytical formulae enable a detailed study of the impact of the two superextreme 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 dragonkings. This allows us to provide a unambiguous distinction between the superextreme dragonkings and ‘normal’ extreme “black swans”. The results illustrate diagnostic that could be useful for the analysis of extreme and superextreme events in real empirical time series. The European Physical Journal Special Topics 205 (2012) 2752. 
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Presentation: Article at Econophysics group of Ryszard Kutner, by Ryszard Kutner Submitted: 20160329 12:43 Revised: 20160329 12:48 