Parameter Estimation Of Stochastic Differential Equations Driven By Fractional Brownian Motion

In recent years,there have been many studies already made on the parameter estimation of stochastic differential equations driven by standard Brownian motion.However,there are few studies on parameter estimation of stochastic differential equa-tions driven by fractional Brownian motion.Fractional Brownian motion is a Gaussian process with self-similarity,non-stationarity and long dependence,which is more suit-able for the actual situation.In this paper,we associate the stochastic process driven by fractional Brownian motion with the semi-martingale.Then the maximum likeli-hood estimation and Bayesian estimation are obtained and the asymptotic properties of the estimators are further proved.This paper consists of four parts.The first chapter mainly introduces the research background and research status in this direction,and the main work of this paper is summarized.The second chapter is preliminary knowledge,which includes some basic concepts of probability theory,random analysis and related theorems needed for later proof.In the third chapter,we briefly describe the transformation of fractional Brown-ian motion into a semi-martingale.After that,the maximum likelihood estimation and Bayesian estimation of the parameter in the generalized Ornstein-Uhlenbeck process driven by fractional Brownian motion are given.We also analyze the strong consis-tency and asymptotic normality of the estimators as time tends to infinity.Chapter fourth considers the Bayesian estimation of unknown parameters in parabolic stochas-tic partial differential equations driven by fractional Brownian motion.By applying the Bernstein-von Mises theorem,the theorem that the Bayesian estimation has the same asymptotic properties as maximum likelihood estimation is proved when the time is fixed and the dimension tends to infinity.