Statistical Inferences For The Ornstein-Uhlenbeck Process And Applications

Ornstein-Uhlenbeck(O-U)type processes,as a class of diffusion processes,play an important role in the fields of finance and physics.In finance,the O-U processes can describe the fluctuation of interest rates and exchange rates,such as the short-term interest rate model-Vasicek model.In physics,they are usually used to simulate the evolution of dynamic systems with stochastic noise.There usually exist some unknown parameters in the above mentioned diffusion processes.In practice,it is necessary to study the precise asymptotic properties of such unknown parameter estimations Therefore,the main aim of this thesis is to investigate the asymptotic properties of the estimators in the O-U process under continuous observation and discrete observations respectively.We can obtain the(self-normalized)Cramér-type moderate deviations and moderate deviations of the log-likelihood ratio process and the unknown parameter estimators.As applications,for a class of hypothesis testing problems,we can show by our theoretical results that the probability of type II error tends to zero exponentially.The main contributions of this thesis include the following two aspects1.For the O-U process in stationary and explosive cases,we study Cramér-type moderate deviations for the log-likelihood ratio process under continuous observations.As an application,we give the negative regions of drift testing problem,and also obtain the decay rates of the error probabilities.The main methods consist of mod-? convergence approach,deviation inequalities for multiple Wiener-It(?) integrals and asymptotic analysis techniques2.Under discrete observations,we study the Cramér-type moderate deviations for parameter estimation in O-U process.Our results contain both stationary and explosive cases.The main methods include the deviation inequalities for multiple Wiener-It(?) integrals,as well as the asymptotic analysis techniques.For applications,we propose test statistics which can be used to construct rejection regions in the hypothesis testing for the drift coefficient,and the corresponding probability of type ?error tends to zero exponentially.