Rubin, Tomas. Frakcionální Brownův Pohyb. 2013. Bachelor’s degree. Charles University in Prague. (written in Czech)
About by Bachelor’s thesis:
I wrote a self-contained presentation of fractional Brownian motion, a stochastic process used in time series analysis for modelling either long-range dependent data (if Hurst parameter H > 0.5) or rough and quickly oscillating data (H < 0.5). Besides the presentation of the theory I extended one qualitative result concerning the random realisation of the process, I described how to simulate the time series, and how to statistically estimate the Hurst parameter.
- First place in the student scientific competition SVOČ, 2013,
- Best Bachelor’s thesis in mathematics at Charles University in Prague, 2013.
Fractional Brownian motion is a nontrivial generalization of standard Brownian motion (Wiener process). Definition leaves independence of increments, whereas dependence is controlled by the Hurst index. This paper deals with proofs of fractional Brownian motion‘s properties such as correlation of increments, selfsimilarity, long-range dependence and analytical properties of its paths, i.e. Hölder continuity and nondifferentiability. Furthermore, the proof of the theorem about nondifferentiability is presented in a stronger form than it is usual in published papers about fractional Brownian motion. Further topics are simulations of the
process’s paths, suitable even for general Gaussian processes, and point estimators of the Hurst index.