Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times

Sebastian, Orzeł and Agnieszka, Wyłomańska (2010): Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times. Forthcoming in:

Abstract

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 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 arithmetic Brownian motion. 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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Subdiffusion, Tempered stable distribution, Calibration
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

[1]Baumer B., Meerschaert M.M., Tempered stable Levy motion and transient super-diffusion, J. Comp. Appl. Math. 233, 2438-2448 (2010).

[2] Borak S., Haerdle W. and Weron R., Stable disstributions, , in: Statistical Tools for Finance and Insurance, P. Cizek, W. Haerdle, R. Weron eds., Berlin, Springer, (2005).

[3]Cadavid A.C., Lawrence J.K. and Ruzmaikin A.A., Anomalous diffusion of solar magnetic elements, Astrophys. J. 521, 844-850 (1999).

[4] Caspi A., Granek R. and Elbaum M., Enhanced diffusion in active intracellular transport, Phys. Rev. Lett. 85, 5655-5658 (2000).

[5] Coffey W., Kalmykov Y.P. and Waldron J.T., The Langevin Equation, World Scientific, Singapore (2004).

[6] Chechkin A.V., Gonchar V.Yu., Klafter J. and Metzler R., Natural cutoff in Levy Flights caused by dissipitive nonlinearity, Phys. Rev E. 72 010101 (2005).

[7] Dubrulle B., Laval J.-Ph., Truncated Levy laws and 2D turbulence, Eur. Phys. J. B. 4, 143-146, (1998).

[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)

[9] Golding I., Cox E.C., Physical nature of bacterial cytoplasm, Phys. Rev. Lett. 96, 098102 (2006).

[10] Gorenflo R., Loutchko J. and Luchko Yu., Computation of the Mittag-Leffer function and its derivatives, Fract. Calc. Appl. Anal. 5 (4), 491-518 (2002).

[11] Hougaard P., A class of multivariate failure time distributions, Biometrika 73, 671-678 (1986).

[12] Janczura J., Wylomanska A., Subdynamics of financial data from fractional Fokker-Planck equation, Acta Phys. Polon. B 40 5, 1341-1351 (2009).

[13] Jha R., Kaw P. K., Kulkarni D.R., Parikh J.C. and Team A., Evidence of Levy stable process in tokamak edge turbulence, Phys. Plasmas 10, 699-704 (2003).

[14] Kim Y.S., Rachev S.T., Bianchi M.L. and Fabozzi F.J., A New Tempered Stable Distribution and Its Application to Finance, Georg Bol, Svetlozar T. Rachev, and Reinold Wuerth (Eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer (2007).

[15] Kim Y.S., Chung D.M., Rachev S.T. and Bianchi M.L., The modified tempered stable distribution, GARCH models and option pricing, Prob. Math. Statist. 29 (1), 91-117 (2009).

[16] Kou S.C., Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins, Ann. Appl. Stat. 2, 501-535 (2008).

[17] Magdziarz M., Weron A. and Weron K., Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation, Phys. Rev. E 75, 016708 (2007).

[18] Magdziarz M., Langevin picture of subdiffusion with infinitely divisible waiting times, J. Stat. Phys. 135, 763-772 (2009).

[19] Magdziarz M., Orzel S. and Weron A., Option pricing in subdiffusive model with infnitely divisible waiting times, submitted (2010).

[20] Meerschaert M.M., Benson D.A., Scheffler H.P., and Baeumer B., Stochastic solution of space-time fractional diffusion equations, Phys. Rev. E 65, 041103 (2002).

[21] Metzler R., Klafter J., The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep.-Rev. 339, 1-77 (2000).

[22] Montroll E.W., Weiss G.H., Random walks on lattices: II, .J. Math. Phys. 6, 167-181 (1965).

[23] Orzel S., Weron A., Calibration of the subdiffusive Black-Scholes model, Acta Phys. Pol. B 41 5, 1051-1059 (2010).

[24] Ott A., Bouchaud J.P., Langevin D. and Urbach W., Anomalous diffusion in "living polymers": A genuine Levy flight?, Phys. Rev. Lett. 65, 2201-2204 (1990).

[25] Pfister G., Scher H., Dispersive (non-Gaussian) transient transport in disordered solids, Adv. Phys. 27, 747-798 (1978).

[26] Platani M., Goldberg I., Lamond A.I., and Swedow J.R., Cajal body dynamics and association with chromatin are ATP-dependent, Nat. Cell Biol. 4, 502-508 (2002).

[27] Rosinski J., Tempering stable processes, Stoch. Proc. Appl. 117, 677-707 (2007).

[28] Scher H., Lax M., Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B 7, 4491-4502 (1973).

[29] Scher H., Montroll E., Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455-2477 (1975).

[30] Sokolov I.M., Chechkin A.V. and Klafter J., Fractional diffusion equation for a power law-truncated Levy process, Pysica A 336, 245-251 (2004)

[31] Stanislavsky A.A., Fractional dynamics from the ordinary Langevin equation, Phys. Rev. E 67, 021111 (2003).

[32] Stanislavsky A.A., Weron K. and Weron A., Diffusion and relaxation controlled by tempered alpha-stable processes, Phys. Rev. E 78 051106 (2008).