Mathew Cater Benavides gave this talk at #UndergraduateSeminar on Fractional Brownian Motion
November 17: Fractional Brownian Motion

Abstract: The classical Brownian motion (or Wiener process) serves as the fundamental object of probability theory with vast theoretical and practical applications in a plethora of fields. In the early 1940’s Kolmogorov sought a natural one parameter extension of the process in aims of modelling turbulence in liquids, the extension consists of retaining the framework of the classical motion by constructing still a continuous centered Gaussian process that retains self similarity (of a now distinct index from that of the classical motion) and stationarity of increments but draws its distinction by parameterizing its specifying covariance structure with what is known as the Hurst index, H∈(0,1), resulting in (for ‘most’ values of H) a non-Markov process allowing it to serve as a popular model for dependent phenomena; this extension has since been kept in common parlance as fractional Brownian motion (fBm). This talk aims to provide discussion and (at times demonstration) of the fundamental properties of the fBm as well as investigate sample path properties’ dependence on the Hurst parameter.