Hurst Exponents, Markov Processes, and Fractional Brownian motion

TL;DR

This paper clarifies the differences between fractional Brownian motion and Gaussian Markov processes with Hurst exponent not equal to 1/2, emphasizing the limitations of using Hurst exponents and one-point densities to infer underlying dynamics.

Contribution

It explicitly distinguishes fBm from Gaussian Markov processes with H ≠ 1/2 and discusses the inadequacy of Hurst exponents and one-point densities for system identification.

Findings

Two-point density of fBm does not scale.

One-point density matches that of certain Markov processes.

Hurst exponents alone are insufficient for deducing underlying dynamics.

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 not equal to 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 is identical with that for a Markov process with H not 1/2. We conclude that both Hurst exponents and histograms for one point densities are inadequate for deducing an underlying stochastic dynamical system from empirical data.