The Study Of Compound Fractional Brownian Motion And Bayesian Estimation Of Hurst Exponent

Fractional Brownian motion(FBM)is very important in the study of long memory process,especially with the development of the stochastic integrals theory and the formation of Black-Scholes option pricing theory,FBM occupies an important position in the time series analysis,and becomes a common mathematical model in natural processes and financial markets,the booming estimator of Hurst exponent also becomes the focus of people's attention.In this paper,the generalized Compound Fractional Brownian motion and a Bayesian estimation method of Hurst exponent is studied.The various properties,simulation methods and application scenarios of FBM are introduced firstly in this paper.Then FBM is generalized to Compound FBM,and it is proved that Compound FBM has similar properties of stationary increment and self-similarity.For current estimation methods of the Hurst exponent has the shortcoming that they can only obtain point estimation,this paper focuses on the study of a Bayesian estimation method of the Hurst exponent in terms of a linear FGN model,then I improved this method by using sliding windows,which improved the unbiasedness and effectiveness of the maximum posterior estimator based on Bayesian method with respect to noisy data.This paper also compares this Bayesian approach with the R/S,Wavelet method with sampling experiments,and proves that the Bayesian approach is the most robust.In addition,this paper discusses the performance of four different window functions in different estimation methods based on artificial noisy FGN data with H <0.5,and found that the Bayesian,R/S method combined with Blackman window,the Wavelet method combined with rectangle window work best.Finally,I obtained time series of the annual water level of the Nile River for the years 1871-1970 from R language dataset,and the time series' long memory can be found obviously with fluctuation analysis,then I used the Wavelet,R/S,Bayesian method combined with rectangular sliding window to estimate the Hurst exponent,the results of three methods are 0.7824,0.8529,0.8462,which are very similar to the results of other researchers.And the variance of maximum a posteriori estimation is the smallest,which verifies the effectiveness and robustness of Bayes estimation method.