Asymptotic Behaviors Of The Trajectory Fitting Estimator In Ornstein-Uhlenbeck Process With Linear Drift

Ornstein-Uhlenbeck(O-U) process plays a crucial role in the research field of Physics and Finance.It is often used to describe the evolutionary process of dynamic systems disturbed by random noise,and the random phenomenon in the theory of Control.As the solution of Langevin equation,O-U process can describe the speed of particle in the Coulonb gas model.Moreover,this process can also be applied to study the fluctuations of interest rate and exchange rate,and as well known,the first short-term interest rate model-Vasicek model is a O-U process with linear drift.However,there always exist some unknown parameters in trend item of O-U process.And then,in order to apply the O-U process to practice,it is necessary to study the asymptotic properties of the estimators of those unknown parameters,which mainly include central limit theorem,moderate and large deviation principle,Berry-Esseen bound,the low of iterated logarithm and its convergence rate.This thesis is organized as follows:In the first chapter,we give the preliminary knowledge,and introduce some existed research results of the maximum likelihood estimator of the trend parameters in O-U process,and write the main results related to the trajectory fitting estimator.In the second chapter,using the It? formula and integration formula by parts,we give the multiple Wiener-It? integral representation for the quadratic functionals in the trajectory fitting estimator.In the third chapter,together with the deviation inequalities for multiple Wiener-It? integral and the technique of separation,we prove the deviation inequalities and moderate deviation principle for the trajectory fitting estimator.In the fourth chapter,applying the multiple Wiener-It? integral representation and the properties of Gauss process,we establish the law of iterated logarithm with its convergence rate for trajectory fitting estimator.In the fifth chapter,the main ideas of the paper are emphasized,and we also analyze the further researching work.