| The primary purpose of this study was to find Bayesian estimates for the Hurst dimension of a fBm with a Beta prior when the process is observed at both discrete and continuous times. Additionally, this study sought to examine how sensitive is the Bayesian analysis with Beta prior to the choice of parameters of Beta prior. Finally, this study attempted to develop R codes for the research questions.;Using Metropolis-Hastings algorithm of MCMC as well as the assumed proposal distribution of Beta distribution, the Bayesian estimate for the Hurst dimension of a fBm with a Beta prior when the process is observed at discrete times was obtained. For the continuous case, however, the probability measures generated by two different Hurst dimension processes are singular with respect to each other, so it follows that there is no likelihood function for the continuous case.;Overall, the estimated H appears to be greater than the real H. Overestimation is observed though the overestimation is less severe as real H goes up. In addition, the estimated H decreases as Beta parameters go up given an Alpha value. In contrast, the estimated H increases as Alpha parameters go up given a Beta value. For the real-world data, the 2011 daily Taiwan Stock Index was used and the estimated Hurst index was 0.21. Finally, the R codes were successfully developed to implement the simulation in this study using a variety of packages such as "dvfBm," "mnormt," and "mcmcse.". |
