The European Physical Journal B

, Volume 66, Issue 1, pp 137–148 | Cite as

On discrete stochastic processes with long-lasting time dependence in the variance

  • S. M.D. Queirós
Interdisciplinary Physics


In this manuscript, we analytically and numerically study statistical properties of an heteroskedastic process based on the celebrated ARCH generator of random variables whose variance is defined by a memory of qm-exponencial, form (eqm=1 x=ex). Specifically, we inspect the self-correlation function of squared random variables as well as the kurtosis. In addition, by numerical procedures, we infer the stationary probability density function of both of the heteroskedastic random variables and the variance, the multiscaling properties, the first-passage times distribution, and the dependence degree. Finally, we introduce an asymmetric variance version of the model that enables us to reproduce the so-called leverage effect in financial markets.


05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 89.90.+n Other topics in areas of applied and interdisciplinary physics 


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  1. J.-P. Bouchaud, M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000); R.N. Mantegna, H.E. Stanley, An introduction to Econophysics: correlations and Complexity in Finance (Cambridge University Press, Cambrigde, 1999); J. Voit, The Statistical Mechanics of Financial Markets (Springer-Verlag, Berlin, 2003); M. Dacorogna, R. Gençay, U. Müller, R. Olsen, O. Pictet, An Introduction to High-Frequency Finance (Academic Press, London, 2001) Google Scholar
  2. S. Campbell, F.X. Diebold, J. Am. Stat. Ass. 100, 6 (2005) Google Scholar
  3. J.D. Martin-Guerrero, G. Camps-Valls, E. Soria-Olivas, A.J. Serrano-Lopez, J.J. Perez-Ruixo, N.V. Jimenez-Torres, IEEE Trans. Biomed. Eng. 50, 1136 (2003) Google Scholar
  4. P. Gronke, J. Brehm, Elect. Stud. 21, 425 (2002) Google Scholar
  5. T.G. Andersen, T. Bollerslev, P.F. Christoffersen, F.X. Diebold, Volatility Forecasting, PIER working paper 05-011, 2005 Google Scholar
  6. R.F. Engle, Econometrica 50, 987 (1982) Google Scholar
  7. S.M. Duarte Queirós, Europhys. Lett. 80, 30005 (2007) Google Scholar
  8. B. Pobodnik, P.C. Ivanov, Y. Lee, A. Cheesa, H.E. Stanley, Europhys. Lett. 50, 711 (2000); S.M. Duarte Queirós, C. Tsallis, Europhys. Lett. 69, 893 (2005) Google Scholar
  9. M. Porto, H.E. Roman, Phys. Rev. E 63, 036128 (2001) Google Scholar
  10. T. Bollerslev, R.Y. Chou, K.F. Kroner, J. Econometrics 52, 5 (1992); T.G. Andersen, T. Bollerslev, F.X. Diebold, in Handbook of Financial Econometrics, edited by Y. Aït-Sahalia (Elsevier, Amsterdam, 2006) Google Scholar
  11. Z. Zing, C.W.J. Granger, R.F. Engle, J. Emp. Fin. 1, 83 (1983) Google Scholar
  12. C. Gourieroux, A. Montfort, Statistics and Econometric Models (Cambridge University Press, Cambridge, 1996) Google Scholar
  13. C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, Braz. J. Phys. 29, 1 (1999) Google Scholar
  14. C.W.J. Granger, Z. Ding, J. Econometrics 73, 61 (1996); H.E. Roman, M. Porto, Int. J. Mod. Phys. C (to be published) Google Scholar
  15. C. Dose, M. Porto, H.E. Roman, Phys. Rev. E 67, 067103 (2003) Google Scholar
  16. G.M. Schütz, S. Trimper, Phys. Rev. E 70, 045101(R) (2004) Google Scholar
  17. Google Scholar
  18. Google Scholar
  19. D.C. Boes, F.A. Graybill, A.M. Mood, Introduction to the Theory of Statistics, 3rd edn. (McGraw-Hill, New York, 1974); H.R. Neave, Statistics Tables for mathematicians, engineers, economists and the behavioural and management sciences (Routledge, London, 1999) Google Scholar
  20. I.S. Gradshteyn, I.M. Ryzhik, Tables of integrals, series and products (Academic Press, London, 1965) Google Scholar
  21. C. Beck, E.G.D. Cohen, Physica A 322, 267 (2003) Google Scholar
  22. A.M. Reynolds, N. Mordant, A.M. Crawford, E. Bodenschatz, New J. Phys. 7, 58 (2005); C. Beck, Phys. Rev. Lett. 98, 064502 (2007) Google Scholar
  23. K. Hlaváčková-Schlinder, M. Paruš, M. Vejmelka, J. Bhattacharya, Phys. Rep. 441, 1 (2007) Google Scholar
  24. C. Tsallis, Phys. Rev. E 58, 1442 (1998) Google Scholar
  25. L. Borland, A.R. Plastino, C. Tsallis, J. Math. Phys. 39, 6490 (1998); L. Borland, A.R. Plastino, C. Tsallis, J. Math. Phys. 40, 2196 (1999) Google Scholar
  26. S.M. Duarte Queirós, Quantitatit. Finance 5, 475 (2005); M. Portesi, F. Pennini, A. Plastino, Physica A 373, 273 (2007); S.M. Duarte Queirós, e-print arXiv:0805.2254 [cond-mat.stat-mech] (preprint, 2008) Google Scholar
  27. S.M.D. Queirós, C. Tsallis, Eur. Phys. J. B 48, 139 (2005) Google Scholar
  28. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer-Verlag, Berlin, 1989) Google Scholar
  29. F. Wang, P. Weber, K. Yamasaki, S. Havlin, H.E. Stanley, Eur. Phys. J. B 55, 123 (2007); E. Scalas, Chaos Solitons & Fractals (to be published) Google Scholar
  30. B. Hoskins, in Predictability of Wheather and Climate, edited by T. Palmer, R. Hagedorn (Cambridge University Press, Cambridge, 2006) Google Scholar
  31. B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman & Co., San Francisco – CA, 1983); J. Feder, Fractals (Plenum, New York, 1988) Google Scholar
  32. B.B. Mandelbrot, Fractals and Scaling in Finance (Springer, New York, 1997) Google Scholar
  33. A. Admati, P. Pfleiderer, Rev. Financial Studies 1 (1988); A. Arnéodo, J.-F. Muzy, D. Sornette, Eur. Phys. J. B 2, 277 (1998), K. Ivanova, M. Ausloos, Eur. Phys. J. 8, 665 (1999); B. Pochart, J.P. Bouchaud, e-print arXiv:cond-mat/0204047 (preprint, 2002); T. Di Matteo, Quantit. Finance 7, 21 (2007); P. Oświęcimka, J. Kwapień, S. Drożdż, Phys. Rev. E 74, 016103 (2006); L.G. Moyano, J. de Souza, S.M. Duarte Queirós, Physica A 371, 118 (2006); F. Wang, K. Yamasaki, S. Havlin, H.E. Stanley, Phys. Rev. E 77, 016109 (2008) Google Scholar
  34. Z. Eisler, J. Kertész, Europhys. Lett. 77, 28001 (2007) Google Scholar
  35. J.P. Bouchaud, M. Potters, M. Meyer, e-print arXiv:cond-mat/9906347 (preprint, 1999); J. de Souza, S.M. Duarte Queirós, e-print arXiv:0711.2550 [] (preprint, 2007); Z.-Q. Jiang, W.-X. Zhou, Physica A 387, 3605 (2008) Google Scholar
  36. J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H.E. Stanley, Physica A 316, 87 (2002) Google Scholar
  37. R.A. Haugen, E. Talmor, W.N. Torous, J. Fin. 46, 985 (1991) Google Scholar
  38. J.P. Bouchaud, A. Matacz, M. Potters, Phys. Rev. Lett. 87, 228701 (2001); J. Masoliver, J. Perelló, Int. J. Theo. Appl. Fin. 5, 541 (2002) Google Scholar
  39. C. Tsallis, Physica A 340, 1 (2004) Google Scholar
  40. Nonextensive Entropy – Interdisciplinary Applications, edited by M. Gell-Mann, C. Tsallis (Oxford University Press, New York, 2004); Complexity, Metastability and Nonextensivity, edited by C. Beck, G. Benedek, A. Rapisarda, C. Tsallis (World Scientific, Singapore, 2005), p. 135; Complexity, Metastability, and Nonextensivity: An International Conference edited by S. Abe, H. Herrmann, P. Quarati, A. Rapisarda, C. Tsallis, AIP Conf. Proc. 965 (2007)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150Rio de JaneiroBrazil

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