Hurst exponents, Markov processes, and nonlinear diffusion equations

Bassler, Kevin E. and Gunaratne, Gemunu H. and McCauley, Joseph L. (2005): Hurst exponents, Markov processes, and nonlinear diffusion equations. Published in: Physica A , Vol. 369, (2006): pp. 343-353.

Abstract

We show by explicit closed form calculations that a Hurst exponent H≠1/2 does not necessarily imply long time correlations like those found in fractional Brownian motion. We construct a large set of scaling solutions of Fokker-Planck partial differential equations where H≠1/2. Thus Markov processes, which by construction have no long time correlations, can have H≠1/2. If a Markov process scales with Hurst exponent H≠ 1/2 then it simply means that the process has nonstationary increments. For the scaling solutions, we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient D(x,t) is specified. As an example, we generate a class of student-t-like densities from the class of quadratic diffusion coefficients. Notably, the Tsallis density is one member of that large class. The Tsallis density is usually thought to result from a nonlinear diffusion equation, but instead we explicitly show that it follows from a Markov process generated by a linear Fokker-Planck equation, and therefore from a corresponding Langevin equation. Having a Tsallis density with H≠1/2 therefore does not imply dynamics with correlated signals, e.g., like those of fractional Brownian motion. A short review of the requirements for fractional Brownian motion is given for clarity, and we explain why the usual simple argument that H≠1/2 implies correlations fails for Markov processes with scaling solutions. Finally, we discuss the question of scaling of the full Green function g(x,t;x',t') of the Fokker-Planck pde.

• Item Type: MPRA Paper
• Institution: University of Houston
• Language: English
• Keywords: Hurst exponent; Markov process; scaling; stochastic calculus; autocorrelations; fractional Brownian motion; Tsallis model; nonlinear diffusion
• Subjects: G - Financial Economics > G1 - General Financial Markets; G - Financial Economics > G1 - General Financial Markets > G10 - General; G - Financial Economics > G1 - General Financial Markets > G14 - Information and Market Efficiency ; Event Studies ; Insider Trading

References

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

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

3. N. Wax. Selected Papers on Noise and Stochastic Processes. Dover: N.Y., 1954.

4. L. Arnold, Stochastic Differential Equations. Krieger, Malabar, Fla., 1992.

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

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

6b. P. Embrechts and M. Maejima, Self-similar Processes, Princeton University Press, Princeton, 2002.

6c. E. Scalas, R. Gorenflo and F. Mainardi, Physica A284, 376, 2000.

6d. F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas Physica A 287, 468, 2000.

7. J. Feder. Fractals. Plenum: N.Y., 1988.

8. L. Borland, Phys. Rev. E57, 6634, 1998.

9. G. Kaniadakis & G. Lapenta, Phys Rev E 62,3246, 2000.

10. G. Kaniadakis & P. Quarati, Physica A237, 229, 1997.

10b. G. Kaniadakis, Physica A 296, 405 (2001).

10c. T. D. Frank, Physica A 331, 391 (2004).

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

12. C.C. Heyde & N.N. Leonenko. Adv. Appl. Prob. 37, 342, 2005.

13. A.M. Yaglom & I.M. Yaglom, An introduction to the Theory of Stationary Random Functions. Transl. and ed. by R. A. Silverman. Prentice-Hall: Englewood Cliffs, N.J., 1962.

14. R. Kubo, M. Toda, & N. Hashitsume. Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer-Verlag: Berlin, 1978.

15.K.Schulten,http://www.ks.uiuc.edu/Services/Class/PHYS498/LectureNotes.html, 1999.

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

17. M. Kac,, Probability and related Topics in Physical Sciences, Wiley-Interscience, N.Y., 1959.

18. A. L. Alejandro-Quinones, K.E. Bassler, J.L. McCauley, and G.H. Gunaratne, in preparation, 2005.

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

20. A. L. Alejandro-Quinones, K.E. Bassler, M. Field, J.L. McCauley, M. Nicol, I. Timofeyev, A. Török, and G.H. Gunaratne, Physica 363A, 383-392, 2006.

20b. A. L. Alejandro-Quinones, K.E. Bassler, J.L. McCauley, and G.H. Gunaratne Proc. of SPIE, Vol. 5848, 27, 2005.

21. J.L. McCauley & G.H. Gunaratne, Physica A329, 178, 2003.

22. B. Mandelbrot and J.R. Wallis, Water Resources Research 4, 909, 1968.

23. R. Mantegna, R. & H.E. Stanley. An Intro. to Econophysics. Cambridge Univ. Pr., Cambridge, 2000.

24. F. Black. J. of Finance 3, 529, 1986.

25. P. Hänggi & H. Thomas, Z. Physik B 26, 85-92, 1977.

25b. R. Hillerbrand & R. Friedrich, Model for the Statistics of Lagrangian Velocity Increments in Fully Developed Turbulence, preprint (2005).

26. J.M. Steele, Stochastic Calculus and Financial Applications. Springer-Verlag, N.Y., 2000.

27. J. L. McCauley, G.H. Gunaratne, & K.E. Bassler, in Dynamics of Complex Interconnected Systems, Networks and Bioprocesses, ed. A.T. Skjeltorp & A. Belyushkin, Springer, NY, 2005. 28. L. Borland, Phy. Rev. Lett. 89, 9, 2002.