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Complex dynamics of financial indices

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Abstract

This paper presents a novel method for the analysis of nonlinear financial and economic systems. The modeling approach integrates the classical concepts of state space representation and time series regression. The analytical and numerical scheme leads to a parameter space representation that constitutes a valid alternative to represent the dynamical behavior. The results reveal that business cycles can be clearly revealed, while the noise effects common in financial indices can elegantly be filtered out of the results.

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References

  1. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Boston (2012)

    Google Scholar 

  2. Bischi, G.I., Chiarella, C., Gardini, L. (eds.): Nonlinear Dynamics in Economics, Finance and the Social Sciences. Springer, Berlin (2010)

    MATH  Google Scholar 

  3. Chen, W.C.: Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals 36, 1305–1314 (2008)

    Article  Google Scholar 

  4. Duarte, F.B., Machado, J.A.T., Duarte, G.M.: Dynamics of the Dow Jones and the NASDAQ stock indexes. Nonlinear Dyn. 61(4), 691–705 (2010)

    Article  MATH  Google Scholar 

  5. Khan, I.R., Ohba, R.: New finite difference formulas for numerical differentiation. J. Comput. Appl. Math. 126(1–2), 269–276 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  7. Li, J.: General explicit difference formulas for numerical differentiation. J. Comput. Appl. Math. 183(1), 29–52 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin, G., Feeny, B.F., Das, T.: Fractional derivative reconstruction of forced oscillators. Nonlinear Dyn. 55(3), 239–250 (2009)

    Article  MATH  Google Scholar 

  9. Ma, J., Bangura, H.I.: Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances. Nonlinear Dyn. 70(4), 2313–2326 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Machado, J.A.T.: Calculation of fractional derivatives of noisy data with genetic algorithms. Nonlinear Dyn. 57(1–2), 253–260 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Machado, J.A.T., Duarte, G.M., Duarte, F.B.: Analysis of financial data series using fractional Fourier transform and multidimensional scaling. Nonlinear Dyn. 65(3), 235–245 (2011)

    Article  Google Scholar 

  12. Machado, J.A.T., Duarte, G.M., Duarte, F.B.: Identifying economic periods and crisis with the multidimensional scaling. Nonlinear Dyn. 63(4), 611–622 (2011)

    Article  Google Scholar 

  13. Machado, J.T., Duarte, F.B., Duarte, G.M.: Analysis of stock market indices through multidimensional scaling. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4610–4618 (2011)

    Article  Google Scholar 

  14. Machado, J.A.T., Duarte, G.M., Duarte, F.B.: Fractional dynamics in financial indexes. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22(10), 1250249 (2012)

    Article  MathSciNet  Google Scholar 

  15. Machado, J.T., Duarte, F.B., Duarte, G.M.: Analysis of financial indices by means of the windowed Fourier transform. Signal Image Video Process. 6(3), 487–494 (2012)

    Article  Google Scholar 

  16. Machado, J.T., Duarte, F.B., Duarte, G.M.: Power law analysis of financial index dynamics. Discrete Dyn. Nat. Soc. 2012, 120518 (2012)

    Google Scholar 

  17. Machado, J.T., Duarte, G.M., Duarte, F.B.: Analysis of stock market indices with multidimensional scaling and wavelets. Math. Probl. Eng. 2012, 819503 (2012)

    Google Scholar 

  18. Meyers, R.A. (ed.): Complex Systems in Finance and Econometrics. Springer, New York (2010)

    Google Scholar 

  19. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  20. Ngounda, E., Patidar, K.C., Pindza, E.: Contour integral method for European options with jumps. Commun. Nonlinear Sci. Numer. Simul. 18(3), 478–492 (2013)

    Article  MathSciNet  Google Scholar 

  21. Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    Google Scholar 

  22. Piqueira, J.R.C., Mortoza, L.P.D.: Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances. Commun. Nonlinear Sci. Numer. Simul. 17(4), 1690–1695 (2012)

    Article  Google Scholar 

  23. Qu, R.: A new approach to numerical differentiation and integration. Math. Comput. Model. 24(10), 55–68 (1996)

    Article  MATH  Google Scholar 

  24. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Amsterdam (1993)

    MATH  Google Scholar 

  25. Silva, S.D., Matsushita, R., Gleria, I., Figueiredo, A., Rathie, P.: International finance, Lévy distributions, and the econophysics of exchange rates. Commun. Nonlinear Sci. Numer. Simul. 10(4), 355–466 (2005)

    Article  MathSciNet  Google Scholar 

  26. Trippi, R.: Chaos & Nonlinear Dynamics in the Financial Markets. Irwin, Burr Ridge (1995)

    Google Scholar 

  27. Vialar, T.: Complex and Chaotic Nonlinear Dynamics. Advances in Economics and Finance, Mathematics and Statistics. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    J. A. Tenreiro Machado

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  1. J. A. Tenreiro Machado
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Correspondence to J. A. Tenreiro Machado.

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Tenreiro Machado, J.A. Complex dynamics of financial indices. Nonlinear Dyn 74, 287–296 (2013). https://doi.org/10.1007/s11071-013-0965-x

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  • DOI: https://doi.org/10.1007/s11071-013-0965-x

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