Open AccessOn some statistics of fractional Brownian motion
Author(s) -
V. М. Bondarenko
Publication year - 2021
Publication title -
sistemnì doslìdžennâ ta ìnformacìjnì tehnologìï
Language(s) - English
Resource type - Journals
eISSN - 2308-8893
pISSN - 1681-6048
DOI - 10.20535/srit.2308-8893.2021.1.11
Subject(s) - fractional brownian motion , mathematics , hurst exponent , brownian excursion , series (stratigraphy) , covariance , brownian motion , limit (mathematics) , mathematical analysis , variance (accounting) , consistency (knowledge bases) , stochastic process , complement (music) , statistics , geometric brownian motion , statistical physics , diffusion process , computer science , discrete mathematics , business , knowledge management , chemistry , innovation diffusion , biology , paleontology , biochemistry , accounting , physics , complementation , gene , phenotype
Fractional Brownian motion as a method for estimating the parameters of a stochastic process by variance and one-step increment covariance is proposed and substantiated. The root-mean-square consistency of the constructed estimates has been proven. The obtained results complement and generalize the consequences of limit theorems for fractional Brownian motion, that have been proved in the number of articles. The necessity to estimate the variance is caused by the absence of a base unit of time and the estimation of the covariance allows one to determine the Hurst exponent. The established results let the known limit theorems to be used to construct goodness-of-fit criteria for the hypothesis “the observed time series is a transformation of fractional Brownian motion” and to estimate the error of optimal forecasting for time series.