Asymptotic Behavior And Ergodicity For The Time Inhomogeneous Diffusion Process

As an important class of stochastic processes,the time inhomogeneous diffusion process is widely used in the financial field.Since the economic conditions always change with time,it is necessary to assume that the instantaneous expected return and instantaneous volatility of financial assets are not only related to the specified state variables,but also depend on time,which means that the state variable is an inhomogeneous diffusion process.However,the diffusion process above often contains unknown parameters.For practical applications,we need to study the asymptotic properties of the estimators of unknown parameters.This thesis will mainly focus on the optimal Berry-Esseen bound,(self-normalized)Cramér-type moderate deviation for the estimator of unknown drift parameter in time inhomogeneous diffusion process.The methods consist of Edgeworth-expansion and Cramér-type moderate deviation of multiple Wiener-It? integrals,Delta method in large deviation theory and some asymptotic analysis techniques.The hypothesis test of ergodicity of the process is also investigated,and the results are applied to the ?-Wiener bridge.The paper is organized as follows.In the first chapter,we briefly describe the research background and significance,main contents and structure of this paper.In the second chapter,we introduce the basic concepts and related results involved in this paper,such as large deviation principle and multiple Wiener-It? integrals.On the basis of a detailed introduction of the model,we introduce the motivation of this paper as well.In the third chapter,we first express the Gaussian functionals as multiple Wiener-It? integrals,then estimate the third and fourth cumulants of multiple Wiener-It? integrals,at last,we study the optimal Berry-Esseen bounds,(self-normalized)Cramér-type moderate deviation of the estimator,by the methods of Edgeworth-expansion and Cramér-type moderate deviation of multiple Wiener-It?integrals.In the fourth chapter,by constructing appropriate statistics and studying their related properties,we test the ergodicity of the process,and establish its unbiasedness and consistency.The last chapter summarizes this thesis and points out some possible research directions in the future.