Calibration of the Subdiffusive Arithmetic Brownian Motion with Tempered Stable Waiting-Times

  • Sebastian Orzeł
  • Agnieszka Wyłomańska


In the classical analysis many models used to real data description are based on the standard Brownian diffusion-type processes. However, some real data exhibit characteristic periods of constant values. In such cases the popular systems seem not to be applicable. Therefore we propose an alternative approach, based on the combination of the popular Brownian motion with drift (called also the arithmetic Brownian motion) and tempered stable subordinator. The probability density function of the proposed model can be described by a Fokker-Planck type equation and therefore it has many similar properties as the popular Brownian motion with drift. In this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic Brownian motion and calibrate the analyzed process to the real financial data.


Subdiffusion Tempered stable distribution Calibration 


  1. 1.
    Baumer, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Borak, S., Haerdle, W., Weron, R.: Stable distributions. In: Cizek, P., Haerdle, W., Weron, R. (eds.) Statistical Tools for Finance and Insurance. Springer, Berlin (2005) Google Scholar
  3. 3.
    Cadavid, A.C., Lawrence, J.K., Ruzmaikin, A.A.: Anomalous diffusion of solar magnetic elements. Astrophys. J. 521, 844–850 (1999) ADSCrossRefGoogle Scholar
  4. 4.
    Caspi, A., Granek, R., Elbaum, M.: Enhanced diffusion in active intracellular transport. Phys. Rev. Lett. 85, 5655–5658 (2000) ADSCrossRefGoogle Scholar
  5. 5.
    Coffey, W., Kalmykov, Y.P., Waldron, J.T.: The Langevin Equation. World Scientific, Singapore (2004) zbMATHGoogle Scholar
  6. 6.
    Chechkin, A.V., Gonchar, V.Yu., Klafter, J., Metzler, R.: Natural cutoff in Lévy flights caused by dissipative nonlinearity. Phys. Rev. E 72, 010101 (2005) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Dubrulle, B., Laval, J.-Ph.: Truncated Levy laws and 2D turbulence. Eur. Phys. J. B 4, 143–146 (1998) ADSCrossRefGoogle Scholar
  8. 8.
    Gajda, J., Magdziarz, M.: Fractional Fokker-Planck equation with tempered alpha-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 82, 011117 (2010) MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Golding, I., Cox, E.C.: Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 098102 (2006) ADSCrossRefGoogle Scholar
  10. 10.
    Gorenflo, R., Loutchko, J., Luchko, Yu.: Computation of the Mittag-Leffler function and its derivatives. Fract. Calc. Appl. Anal. 5(4), 491–518 (2002) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hougaard, P.: A class of multivariate failure time distributions. Biometrika 73, 671–678 (1986) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Janczura, J., Wyłomańska, A.: Subdynamics of financial data from fractional Fokker-Planck equation. Acta Phys. Pol. B 40(5), 1341–1351 (2009) ADSGoogle Scholar
  13. 13.
    Janczura, J., Orzeł, S., Wyłomańska, A.: Subordinated α-stable Ornstein–Uhlenbeck process as a tool of financial data description (2011, submitted) Google Scholar
  14. 14.
    Jha, R., Kaw, P.K., Kulkarni, D.R., Parikh, J.C., Team, A.: Evidence of Lévy stable process in tokamak edge turbulence. Phys. Plasmas 10, 699–704 (2003) ADSCrossRefGoogle Scholar
  15. 15.
    Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J.: A new tempered stable distribution and its application to finance. In: Bol, G., Rachev, S.T., Wuerth, R. (eds.) Risk Assessment: Decisions in Banking and Finance. Physika-Verlag/Springer, Heidelberg (2007) Google Scholar
  16. 16.
    Kim, Y.S., Chung, D.M., Rachev, S.T., Bianchi, M.L.: The modified tempered stable distribution, GARCH models and option pricing. Probab. Math. Stat. 29(1), 91–117 (2009) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kou, S.C.: Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. Ann. Appl. Stat. 2, 501–535 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Magdziarz, M.: Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135, 763–772 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75, 016708 (2007) ADSCrossRefGoogle Scholar
  20. 20.
    Magdziarz, M., Orzeł, S., Weron, A.: Option pricing in subdiffusive model with infinitely divisible waiting times (2010, submitted) Google Scholar
  21. 21.
    Meerschaert, M.M., Benson, D.A., Scheffler, H.P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 041103 (2002) MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep.-Rev. 339, 1–77 (2000) MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Montroll, E.W., Weiss, G.H.: Random walks on lattices: II. J. Math. Phys. 6, 167–181 (1965) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Orzeł, S., Weron, A.: Calibration of the subdiffusive Black-Scholes model. Acta Phys. Pol. B 41(5), 1051–1059 (2010) Google Scholar
  25. 25.
    Ott, A., Bouchaud, J.P., Langevin, D., Urbach, W.: Anomalous diffusion in “living polymers”: a genuine Levy flight? Phys. Rev. Lett. 65, 2201–2204 (1990) ADSCrossRefGoogle Scholar
  26. 26.
    Pfister, G., Scher, H.: Dispersive (non-Gaussian) transient transport in disordered solids. Adv. Phys. 27, 747–798 (1978) ADSCrossRefGoogle Scholar
  27. 27.
    Platani, M., Goldberg, I., Lamond, A.I., Swedow, J.R.: Cajal body dynamics and association with chromatin are ATP-dependent. Nat. Cell Biol. 4, 502–508 (2002) CrossRefGoogle Scholar
  28. 28.
    Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117, 677–707 (2007) zbMATHCrossRefGoogle Scholar
  29. 29.
    Scher, H., Lax, M.: Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7, 4491–4502 (1973) MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Scher, H., Montroll, E.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455–2477 (1975) ADSCrossRefGoogle Scholar
  31. 31.
    Sokolov, I.M., Chechkin, A.V., Klafter, J.: Fractional diffusion equation for a power law-truncated Lévy process. Physica A 336, 245–251 (2004) ADSCrossRefGoogle Scholar
  32. 32.
    Stanislavsky, A.A.: Fractional dynamics from the ordinary Langevin equation. Phys. Rev. E 67, 021111 (2003) MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Stanislavsky, A.A., Weron, K., Weron, A.: Diffusion and relaxation controlled by tempered α-stable processes. Phys. Rev. E 78, 051106 (2008) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer Science, Hugo Steinhaus CenterWrocław University of TechnologyWrocławPoland

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