Subdynamics of financial data from fractional Fokker-Planck equation

Janczura, Joanna and Wyłomańska, Agnieszka (2009): Subdynamics of financial data from fractional Fokker-Planck equation. Published in: Acta Physica Polonica B , Vol. 40, No. 5 (2009): pp. 1341-1351.

Abstract

In exhibition of many real market data we observe characteristic traps. This behavior is especially noticeable for processes corresponding to stock prices. Till now, such economic systems were analyzed in the following manner: before the further investigation trap-data were removed or omitted and then the conventional methods used. Unfortunately, for many observations this approach seems not to be reasonable therefore we propose an alternative approach based on the subdiffusion models that demonstrate such characteristic behavior and their corresponding probability density function (pdf) is described by the fractional Fokker-Planck equation. In this paper we model market data using subdiffusion with a constant force. We demonstrate properties of the considered systems and propose estimation methods.

• Item Type: MPRA Paper
• Language: English
• Keywords: subdiffusion, constant periods, fractional Fokker-Planck equation, stock prices
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation

References

[1] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000)

[2] S. Jespersen, R. Metzler and H. C. Fogedby, Phys. Rev. E 59, 2736 (1999)

[3] B. Szabat, K. Weron and P. Hetman, Phys. Rev. E 75, 021114 (2007)

[4] S. G. Samko, A. A. Kilbas and D. I. Maritchev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Gordon and Breach, Amsterdam (1993)

[5] A. Jurlewicz, A. Wylmanska, P. Zebrowski, Acta Phys. Pol. A 114(3), 629-634 (2008)

[6] M.M. Meeschaert and E. Scalas, Physica A 370, 114 (2006)

[7] E. Broszkiewicz-Suwaj and A. Jurlewicz, Physica A 385, 5503 (2008)

[8] M. Magdziarz, A. Weron and K. Weron, Phys. Rev. E 77, 036704 (2008)

[9] I. M. Sokolov and J. Klafter, Phys. Rev. Lett. 97, 140602 (2006)

[10] M. Magdziarz, A. Weron, K. Weron, Phys. Rev. E 75, 016708 (2007)

[11] M. Magdziarz and A. Weron, Phys. Rev. E 75, 056702 (2007)

[12] M. Magdziarz and A. Weron, Phys. Rev. E 76, 066708 (2007)

[13] D. Fulger, E. Scalas and G. Germano, Phys. Rev. E 77, 021122 (2007)

[14] A. Piryantiska, A. I. Saichev and W. A. Woyczynski, Physica A 349, 375 (2005)

[15] M. Magdziarz and K. Weron, Physica A 376, 1 (2006)

[16] A. Janicki and A. Weron, Simulation and Chaotic Behaviour of Alpha-Stable Stochastic Processes, Mercel Dekker, New York (1994)

[17] A. N. Lagera, J. Appl. Probab. 42, 1134 (2005)

[18] M. Magdziarz, J. Stat. Phys. 136, 553–564 (2009)

[19] M.M. Meeschaert and H. Scheffler, J. Appl. Prob. 41, 623 (2004)

[20] M. Magdziarz, Stoch. Proc. Appl. 119, 3238–3252 (2009)

[21] M. B. Hill, Ann. Statist. 3, 1163 (1975)

[22] R. Weron, Int. J. Mod. Phys. C 12(2), 209 (2001)

[23] J. Pickands Ann. Statist. 3 119 (1975)

[24] A. L. M. Dekkers, J. H. J. Einmahl, L. de Haan, Ann. Statist. 17, 1833 (1989)

[25] A. A. Balkema, L. de Haan, Ann. Prob. 2(5), 792 (1974)

[26] M.M. Meeschaert and H. Scheffler, J. Stat. Plan. Infer. 71, 19 (1998)

[27] http://www.baltic.nordicexchange.com

[28] http://www.gpw.pl

[29] A. Kabasinskas, S. T. Rachev, L. Sakalauskas, W. Sun and I. Belovas, ”Alpha-stable paradigm in financial markets”, technical report (2008)