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Hurst exponents, Markov processes, and fractional Brownian motion

Author

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

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’.

Suggested Citation

  • McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2006. "Hurst exponents, Markov processes, and fractional Brownian motion," MPRA Paper 2154, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:2154
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    File URL: https://mpra.ub.uni-muenchen.de/2154/1/MPRA_paper_2154.pdf
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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.

    More about this item

    Keywords

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

    JEL classification:

    • G00 - Financial Economics - - General - - - General
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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