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By Philipp Frank, Richard Von Mises

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18. A. W. Lo and A. C. MacKinlay, A Non-Random Walk Down Wall Street. (Princeton University Press, 2001). 19. T. Di Matteo, Multi-scaling in finance, Quantitative Finance. 7, 21–36, (2007). 20. T. Di Matteo, T. Aste, and M. M. Dacorogna, Scaling behaviors in differently developed markets, Physica A. , (2003). 21. T. Di Matteo, T. Aste, and M. M. Dacorogna, Long term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development, Journal of Banking & Finance.

Such a difference is reflected in the diffusion dynamics of the random walker. 1. Super-diffusive processes In additive stochastic processes [such as Eq. e. p(η) ∼ |η|−α−1 , with 0 < α < 2), all the motion is dominated by the large fluctuations and it results in a super-diffusive behavior where the mean January 6, 2010 22 17:1 World Scientific Review Volume - 9in x 6in SS22˙Master T. Aste and T. Di Matteo square displacement increases faster than τ . Let us here give an heuristic simple derivation of this fact which shows clearly the origin of anomalous diffusion behavior in presence of power law noise.

I. M. Langlands and P. 1 Diffusion on fractals . . . . . . . . . . 2 Fractional Brownian motion . . . . . . . . 3 Continuous time random walks and power laws . . 4 Simulating random walks for fractional diffusion . . 5 Fractional Fokker–Planck equations . . . . . . 6 Fractional Reaction-Diffusion equations . . . . . 7 Fractional diffusion based models . . . . . . 8 Power laws and fractional diffusion . . . . . . 3 Appendix: Introduction to fractional calculus .

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