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Power-law distributions in economics: a nonextensive statistical approach

Author

Listed:
  • Silvio M. Duarte Queiros
  • Celia Anteneodo
  • Constantino Tsallis

Abstract

The cornerstone of Boltzmann-Gibbs ($BG$) statistical mechanics is the Boltzmann-Gibbs-Jaynes-Shannon entropy $S_{BG} \equiv -k\int dx f(x)\ln f(x)$, where $k$ is a positive constant and $f(x)$ a probability density function. This theory has exibited, along more than one century, great success in the treatment of systems where short spatio/temporal correlations dominate. There are, however, anomalous natural and artificial systems that violate the basic requirements for its applicability. Different physical entropies, other than the standard one, appear to be necessary in order to satisfactorily deal with such anomalies. One of such entropies is $S_q \equiv k (1-\int dx [f(x)]^q)/(1-q)$ (with $S_1=S_{BG}$), where the entropic index $q$ is a real parameter. It has been proposed as the basis for a generalization, referred to as {\it nonextensive statistical mechanics}, of the $BG$ theory. $S_q$ shares with $S_{BG}$ four remarkable properties, namely {\it concavity} ($\forall q>0$), {\it Lesche-stability} ($\forall q>0$), {\it finiteness of the entropy production per unit time} ($q \in \Re$), and {\it additivity} (for at least a compact support of $q$ including $q=1$). The simultaneous validity of these properties suggests that $S_q$ is appropriate for bridging, at a macroscopic level, with classical thermodynamics itself. In the same natural way that exponential probability functions arise in the standard context,power-law tailed distributions, even with exponents {\it out} of the L\'evy range, arise in the nonextensive framework. In this review, we intend to show that many processes of interest in economy, for which fat-tailed probability functions are empirically observed, can be described in terms of the statistical mechanisms that underly the nonextensive theory.

Suggested Citation

  • Silvio M. Duarte Queiros & Celia Anteneodo & Constantino Tsallis, 2005. "Power-law distributions in economics: a nonextensive statistical approach," Papers physics/0503024, arXiv.org.
  • Handle: RePEc:arx:papers:physics/0503024
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    Cited by:

    1. Urbanowicz, Krzysztof & Richmond, Peter & Hołyst, Janusz A., 2007. "Risk evaluation with enhanced covariance matrix," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 468-474.
    2. Kaizoji, Taisei, 2006. "An interacting-agent model of financial markets from the viewpoint of nonextensive statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 109-113.
    3. Lasko Basnarkov & Viktor Stojkoski & Zoran Utkovski & Ljupco Kocarev, 2019. "Option Pricing With Heavy-Tailed Distributions Of Logarithmic Returns," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-35, November.
    4. Sabrina Camargo & Silvio M. Duarte Queiros & Celia Anteneodo, 2013. "Bridging stylized facts in finance and data non-stationarities," Papers 1302.3197, arXiv.org, revised May 2013.
    5. de Mattos Neto, Paulo S.G. & Silva, David A. & Ferreira, Tiago A.E. & Cavalcanti, George D.C., 2011. "Market volatility modeling for short time window," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3444-3453.
    6. Lima, Leonardo S. & Santos, Greicy K.C., 2018. "Stochastic process with multiplicative structure for the dynamic behavior of the financial market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 222-229.
    7. Gontis, V. & Ruseckas, J. & Kononovičius, A., 2010. "A long-range memory stochastic model of the return in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(1), pages 100-106.

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