Hurst exponents, Markov processes, and fractional Brownian motion

McCauley, Joseph L. and Gunaratne, Gemunu H. and Bassler, Kevin E. (2006): Hurst exponents, Markov processes, and fractional Brownian motion. Forthcoming in: Physica A (2007)

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’.

• Item Type: MPRA Paper
• Institution: University of Houston
• Language: English
• Keywords: Markov processes; fractional Brownian motion; scaling; Hurst exponents; stationary and nonstationary increments; autocorrelations
• Subjects: G - Financial Economics > G0 - General > G00 - General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General

References

1. P. Embrechts and M. Maejima, Self-similar Processes, Princeton University Press, Princeton, 2002.

2.B. Mandelbrot & J. W. van Ness, SIAM Rev. 10, 2, 422,1968.

3. B. Mandelbrot, Fractals and Scaling in Finance, Springer, N.Y., 1997.

4. J.L. McCauley, Dynamics of Markets: Econophysics and Finance, Cambridge, Cambridge, 2004.

5. T. Di Matteo, T.Aste, & M.M. Dacorogna, Physica A324, 183, 2003.

6. K.E. Bassler, G.H. Gunaratne, & J. L. McCauley, Hurst Exponents, Markov Processes, and Nonlinear Diffusion Equations, Physica A369, 343, 2006.

7. K.E. Bassler, J. L. McCauley, & G.H. Gunaratne, Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets, 2006.

8. E. Scalas, R. Gorenflo and F. Mainardi, Physica A284, 376, 2000.

9. F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas Physica A 287, 468, 2000.

10. R.L. Stratonovich. Topics in the Theory of Random Noise, Gordon & Breach: N.Y., tr. R. A. Silverman, 1963.

11.See the article by Wang & Uhlenbeck in N. Wax. Selected Papers on Noise and Stochastic Processes. Dover: N.Y., 1954.

12. R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, 1984.

13. G.H. Gunaratne & J.L. McCauley. Proc. of SPIE conf. on Noise & Fluctuations 2005, 5848,131, 2005.

14. A. L. Alejandro-Quinones, K.E. Bassler, M. Field, J.L. McCauley, M. Nicol, I. Timofeyev, A. Török, and G.H. Gunaratne, Physica 363A, 383, 2006.

15. P. Hänggi & H. Thomas, Physics Reports 88, 207, 1982.

16. L. Borland, Quantitative Finance 2, 415, 2002.