Bayesian parameter estimation of the hurst index of fractional brownian motion

Abstract

One of the many generalizations of Brownian Motion, Fractional Brownian Motion is very popular due to being able to account for a wide range of phenomena in various different fields ranging from finance when modelling stock data to hydrology in water turbidity analysis. Brownian Motion is unsuitable for modelling these due to the assumption of independence of increments, an assumption relaxed by Fractional Brownian Motion given it allows dependence of increments. This dependence is effected through the Hurst Index H, the parameter associated with the process. For values of H between 0 and 0.5, both excluded, negative autocorrelation between the increments is enforced while a positive one is obtained for values between 0.5 and 1, both excluded. As H approaches 0.5, the paths of the process will resemble those given by Brownian Motion and if equality holds, the process reduces to a Brownian Motion Process. In this thesis, the theory behind Fractional Brownian Motion as well as the Bayesian framework in the context of estimating H shall be discussed. Given the subjectivity involved in determining a prior, sensitivity analysis shall be performed as to analyze the effect of different priors on the posterior. Following this, the Hurst Index of real data shall be estimated. The data considered shall be genetic data relating to Covid-19 and cardiology data relating to heart rate variation.