Eur. Phys. J. Special Topics 205, 27-52 (2012)
https://doi.org/10.1140/epjst/e2012-01560-0

Regular Article

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

¹ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Hoża 69, 00681 Warsaw, Poland
² Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Hoża 69, 00681 Warsaw, Poland
³ ETH-Zürich, Department of Management, Technology and Economics, Kreuzplaz 5, 8032 Zürich, Switzerland

Received: 23 November 2011
Revised: 9 March 2012
Published online: 1 May 2012

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.