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Dynamics of the Warsaw Stock Exchange index as analysed by the fractional relaxation equation 
Marzena Kozłowska , Ryszard Kutner 
Warsaw University, Faculty of Physics, Hoża 69, Warszawa 00681, Poland 
Abstract 
We studied the historical Warsaw Stock Exchange (WSE) index (WIG) at a daily time horizon; we expect that its dynamics is a typical for an emerging financial market of moderate size We found that the well developed local maxima of the index (except the left side of the first one) can be fitted (up to its fluctuations) by an intermediate part of the MittagLeffler (ML) function which is a natural generalisation of the exponential one. In our case we obtained the exponent α defining the ML function as α < 0.5 for WIG's maxima; the left side of the first maximum is well fitted by the exponential function. Note that the ML function has two characteristic limits: (i) for the initial time the streched exponential form or KohlrauschWilliamsWatts (KWW) and (ii) for the asymptotic time the powerlaw or the Nuttig law. These decays are typical for the relaxation of disordered systems, e.g. viscoelastic materials, and lead to the ColeCole behaviour for the complex susceptibility which is broadely used to describe empirical data. The ML function can be applied to describe dynamics in both time directions. In other words, the relaxation of the WIG local maxima can be described by the fractional (nonDebye) relaxation equation which has indeed the solution in the form of the ML function. Since we found that most of empirical WIG maxima are well covered by the intermediate part of the ML function, this means that the WSE is a complex system lying between two different types of disordered ones created by stock market investors and described, correspondingly, by two types of relaxation functions (i) and (ii). Unfortunately, this observation does not uniquely define the microeconomical (or microscopic) model which constitutes its basis. 
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Presentation: Oral at 2 Ogólnopolskie Sympozjum "Fizyka w Ekonomii i Naukach Społecznych", Plenary session, by Marzena Kozłowska Submitted: 20051116 17:03 Revised: 20090607 00:44 