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. 343353.

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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 FokkerPlanck 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 studenttlike 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 FokkerPlanck 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 FokkerPlanck pde.
Item Type:  MPRA Paper 

Institution:  University of Houston 
Original Title:  Hurst exponents, Markov processes, and nonlinear diffusion equations 
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 
Item ID:  2152 
Depositing User:  Joseph L. McCauley 
Date Deposited:  09. Mar 2007 
Last Modified:  17. Feb 2013 17:07 
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URI:  http://mpra.ub.unimuenchen.de/id/eprint/2152 