Peculiarities of estimating the Hurst exponent of classical Brownian motion, using the R/S Analysis

Abstract

The features of Hurst exponent H of the classical Brownian motion trajectory calculated by the R/S-analysis has been studied, where R is a range of a cumulative deviations of the chosen fragment of the trajectory within the time interval  (from the mathematical aspect – time series (TS)), S is a mathematical expectation of the fragment of the analyzed TS. Due to the fact that while calculating the estimates of Hurst Exponent H of the analyzed TS using the R/S analysis, it is required to set up parameters values , the assumption was made on the effect of these parameters on the estimate of Hurst Exponent H. During the confirmation of the suggested hypothesis it was found that the estimates of the Hurst exponent H coincided with the accuracy up to the calculations error with the original Hurst Exponent of the classical Brownian motion equal to 0.5, only for the particular pairs of values , . It is shown that on the plane pairs of the values  are located along the line  When the arbitrary choice of the R/S method parameters ensures _k, , Hurst exponent varies in the span [0.25; 1.12].

The observed characteristic feature of the estimates of Hurst Exponent H by the R/S analysis of the classical Brownian motion makes it possible to suggest similar features of the estimates of Hurst Exponent H by the R/S analysis of the Fractional Brownian motion, if the suggestion proves to be true, it will be necessary to conduct a critical analysis of the results of a great number of publications where the authors used an R/S method.