Hurst exponents, Markov processes, and nonlinear diffusion equations

Kevin E. Bassler, Gemunu H. Gunaratne and Joseph L. McCauley

MPRA Paper from University Library of Munich, Germany

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.

Keywords: Hurst exponent; Markov process; scaling; stochastic calculus; autocorrelations; fractional Brownian motion; Tsallis model; nonlinear diffusion (search for similar items in EconPapers)
JEL-codes: G1 G10 G14 (search for similar items in EconPapers)
Date: 2005-12-01
New Economics Papers: this item is included in nep-ecm and nep-ets
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://mpra.ub.uni-muenchen.de/2152/1/MPRA_paper_2152.pdf original version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:pra:mprapa:2152

Access Statistics for this paper

More papers in MPRA Paper from University Library of Munich, Germany Ludwigstraße 33, D-80539 Munich, Germany. Contact information at EDIRC.
Bibliographic data for series maintained by Joachim Winter (winter@lmu.de).