Fraction Of O-u Process Of Bayesian Analysis And Its Applications

In this thesis, we are concerned with the problem of theBayesian Estimation of parameters from Fractional Ornstein-Uhle nbeck process, which is driven by a Fractional Brownian noise Generally speaking, classical Ornstein-Uhlenbeck (O-U for abbreviation) processes are Gauss- Markov procession, which makes the parameter estimation much more easy, and we can manage it just following the basic steps of bayesian analysis. However, when a Fractional Brownian motion, instead of a standard one, drives an O-U process the problem becomes complexity. First of all, we have to give estimation to another new parameter, which is called Hurst index (See the definition in section two), Secondly, the process is no longer a martingale, but a long-term dependent process. With these changes, we have to overcome many difficulties, for instance, the calculation of the likelihood function, the integral of the poster distribution, est. In order to obtain our target, we choose to apply the Monte Carlo Markov Chain (MCMC) method, and the Gibbs sampling strategy is used. Meanwhile, new problems appeared during the execution of this method: With the complexity of the target function, it takes too long to draw a sample from the distribution in the usual way, so, how to improve the sampling efficiency? And as the needs of MCMC method, we have to decide when shall we stop the sampling, how to diagnose the convergence of the sample chain? And how to perform the model criterion? In our work we have successfully overcome above problems ,and obtained the bayesian estimation of the parameters, especially the Hurst index, we find that theShanghai Stock 180 index shows a strong long term dependence , with a Hurst index about 0.6 and we also gave a simple research over the distribution of the returns of Shanghai index ,by simulate the index with a certain distribution , which is a distribution family includes normal distribution as a special case,and it is supposed to have thick tail and high peaks, just as the data shows. And the Maximum Likelihood Estimation of the parameters are obtained. Only to find that such distribution can describe the data reasonably.