### 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.

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B.B. Mandelbront & J.W. Van Ness. Fractional Brownian Motions, Fractional Noise and Applications // SIAM Review, Vol. 10, № 4, 1968. P. 422-437.

Kolmogorov A.N. Spiral' Vinera i nekotorye drugie interesnye krivye v gil'bertovom prostranstve Doklady Akademii nauk SSSR, 1940. Т. 26, № 1, с. 115-118. (in Russian)

R. Crownover, Introduction to Fractals and Chaos, Jones & Barlett Publishers, 1995. 306 с.

H. E. Hurst, Long-term storage capacity of reservoirs // Transactions of the American Society of Civil Engineers, 1951, Vol. 116, Issue 1, p. 770-799

E. Peters, Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility John Wiley & Sons, 1996. 288 с.

Zinenko A.V. R/S-analiz na fondovom rynke / /Biznes-informatika, Zinenko A.V. R/S-analiz na fondovom rynke / /Biznes-informatika, 2012. № 3(21). С. 24 -30. (in Russian)

SHeluhin O.I. Samopodobie i fraktaly: telekommunikacionnye prilozheniya, O.I. SHeluhin, A.V. Osin, S.M. Smol'skij, M: Fizmatlit, 2008. 368 с. (in Russian)

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