Fractional Brownian Motion: Problems of Extrapolation and Point Estimation

The article is dedicated to some special properties and characteristics of fractional Brownian motion random process and wider set of self-similar random processes with stationary increments with parameter H. Fractional Brownian motion (further - Fbm) is widely used in forecasting, trend estimations, turbulence fields modelling etc. It was first considered by British climate scientist Harold Hurst in his work published in 1951 and focused on Nil river research. Improved Fbm models were also integrated into the financial analysis of different stationary indicators.
According to it, two problems categories were considered: extrapolation problem (forecasting values for a finite time) and point estimation of Hurst parameter H in Fbm model. Moreover, some new results were achieved for all self-similar random processes with stationary increments (we’ll call them fractional further). More precisely:
-integral and spectral representation for fractional processes was found;
-an explicit form of linear extrapolation functional and square mean error of the extrapolation was calculated;
-some sample statistics were considered (sample mean, autocovariance and autocorrelation) concerning the problem of point estimation. Particularly, convergence almost surely was proved (all limits were also calculated), and some limit theorems related to sample statistics were proved for all H values.
A new estimation of the Hurst parameter was suggested with the research of its asymptotic behaviour. It was also compared with some other well-known estimations.
As a result, the problem of extrapolation was solved for all fractional processes, including Fbm. A new estimation of the Hurst parameter was suggested, some asymptotic properties were also researched using limit theorems for sample statistics.