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Anomalous leftsided multifractal structure of intertransation timeintervals and the possible thirdorder phase transition on financial market 
Andrzej Kasprzak ^{1}, Josep Perelló ^{2}, Jaume Masoliver ^{2}, Ryszard Kutner ^{1} 
1. Warsaw University, Faculty of Physics, Hoża 69, Warszawa 00681, Poland 
Abstract 
We extended the ContinuousTime Random Walk (CTRW) formalism to de scribe the anomalous multifractal structure of random intertransaction timeintervals, which we found when e.g., futures on USD/DM foreign exchange rate (on FX market, archival data) were intensively traded within as many as it was possible time scales. We found that within this extended formalism the scaling powerdependent partition function, Z(q), diverges for any negative scaling powers q (which justify the name anomalous) while for the positive ones it possesses the general scaling with exponent τ(q). In the definition of the partition function we used the PausingTime Distribution (PTD) as the central one, which has the form of the convolution (or superstatistics used e.g., for description of turbulence as well as financial market). Its integral kernel is given by the stretched exponential function (often used in disordered systems) as a generalisation of the exponential distribution assumed in the original version of the CTRW formalism (for description of the transient photo current measured in amorphous glossy material) and the Gaussian one sometimes used in this context (e.g., for diffusion of hydrogen in amorphous metals); more rafined but heuristic analytical prediction was also considered. We argued that this superstatistics define, in fact, a more general multiplicative multifractal structure based on a generalised multiplicative cascadic process (while geometrical multiplicative cascadic process was used e.g., in the fully developed turbulence). The most important results which we found by using the SaddlePoint Approximation are as follows: (i) the spectrum of singularities is unusual namely, it is the leftsided and unlimited for its right side and (ii) the thirdorder phase transition was found which can be roughly interpreted as transition between the phase where high frequency trading is most visible and the phase defined by the low frequency trading. This is the first time when both considered properties were simultaneously observed on financial market although, limited leftsided multifractals were already introduced and considered by Mandelbrot and Evertsz while in the context of phase transition in DLA the limited leftsided multifractal was found by Stanley. 
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Presentation: Poster at International Conference on Economic Science with Heterogeneous Interacting Agents 2008, by Andrzej Kasprzak Submitted: 20080510 18:36 Revised: 20090607 00:48 