Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems |
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Authors: | ZF Huang S Solomon |
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Institution: | Institute for Theoretical Physics, Cologne University, 50923 K?ln, Germany, DE Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel, IL
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Abstract: | We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic
systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the
size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on
the way the elements with very small values are treated. These truncated power laws determine the collective time evolution
of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according
to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations
are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential
decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets
and to the exponential cut-off detected in certain recent measurement.
Received 29 July 2000 and Received in final form 25 September 2000 |
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Keywords: | PACS 05 40 +j Fluctuation phenomena random processes noise and Brownian motion – 05 70 Ln Nonequilibrium and irreversible thermodynamics – 02 50 -r Probability theory stochastic processes and statistics |
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