- fractional calculus modelling


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December 2000 - 001201
Fractional Calculus in Econophysics

Selected papers from the International Workshop Economics Dynamics from the Physics Point of View, Physikzentrum Bad Honnef, Germany 27 - 30 March 2000 have been published in the volume 287, Nos. 3-4 (1 december 2000) of Pysica A - Statistical Mechanics and its Applications (

We point out two papers on the application of Fractional Calculus in Econophysics:
F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas: Fractional calculus and continuous-time finance II: the waiting-time distribution, pp. 468-481.

ABSTRACT:    We complement the theory of tick-by-tick dynamics of financial markets based on a continuous-time random walk (CTRW) model recently proposed by Scalas et al. (Physica A 284 (2000) 376), and we point out its consistency with the behaviour observed in the waiting-time distribution for BUND future prices traded at LIFFE, London.

PACS: 02.50.r; 02.50.Ey; 02.50.Wp; 89.90.+n

Keywords: Stochastic processes; Continuous-time random walk; Fractional calculus; Statistical finance; Econophysics

N. Laskin: Fractional market dynamics, pp. 482-492.

ABSTRACT:    A new extension of a fractality concept in financial mathematics has been developed. We have introduced a new fractional Langevin-type stochastic differential equation that differs from the standard Langevin equation: (i) by replacing the first-order derivative with respect to time by the fractional derivative of order Á; and (ii) by replacing "white noise" Gaussian stochastic force by the generalized "shot noise", each pulse of which has a random amplitude with the alpha-stable LÚvy distribution. As an application of the developed fractional non-Gaussian dynamical approach the expression for the probability distribution function (pdf) of the returns has been established. It is shown that the obtained fractional pdf fits well the central part and the tails of the empirical distribution of S&P 500 returns.

PACS: 02.50.Ey; 05.45.Df; 05.40.Fb

Keywords: Fractal; Shot noise; LÚvy alpha-stable process;