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Hurst exponents, Markov processes, and fractional Brownian motion

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  • McCauley, Joseph L.
  • Gunaratne, Gemunu H.
  • Bassler, Kevin E.
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    Abstract

    There is much confusion in the literature over Hurst exponents. Recently, we took a step in the direction of eliminating some of the confusion. One purpose of this paper is to illustrate the difference between fBm on the one hand and Gaussian Markov processes where H≠1/2 on the other. The difference lies in the increments, which are stationary and correlated in one case and nonstationary and uncorrelated in the other. The two- and one-point densities of fBm are constructed explicitly. The two-point density doesn’t scale. The one-point density for a semi-infinite time interval is identical to that for a scaling Gaussian Markov process with H≠1/2 over a finite time interval. We conclude that both Hurst exponents and one point densities are inadequate for deducing the underlying dynamics from empirical data. We apply these conclusions in the end to make a focused statement about ‘nonlinear diffusion’.

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    Bibliographic Info

    Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 2154.

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    Date of creation: 30 Sep 2006
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    Handle: RePEc:pra:mprapa:2154

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    Related research

    Keywords: Markov processes; fractional Brownian motion; scaling; Hurst exponents; stationary and nonstationary increments; autocorrelations;

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    Cited by:
    1. McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2008. "Martingales, nonstationary increments, and the efficient market hypothesis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3916-3920.
    2. Bassler, Kevin E. & Gunaratne, Gemunu H. & McCauley, Joseph L., 2008. "Empirically based modeling in financial economics and beyond, and spurious stylized facts," International Review of Financial Analysis, Elsevier, vol. 17(5), pages 767-783, December.

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