Parameter Estimation For Several Classes Of It(?) Stochastic Differential Equations

This thesis is concerned with the problem of parameter estimation for diffusion processes defined by It(?) stochastic differential equations. The content of this thesis is mainly divided into four parts. In the first part, the existence and consistency of the parameter estimator and asymptotic normality of the error of estimation for a class of continuoustime ergodic diffusion processes under the condition of continuoustime observations are discussed. In the second part, the consistency of the parameter estimator and asymptotic normality of the error of estimation for a class of continuoustime diffusion processes are investigated form discrete observations. In the third part, the consistency of the parameter estimator and asymptotic normality of the error of estimation for two widely used economic models called CIR and CI are studied, the hypothesis testing is made to verify the effectiveness of the estimation approach. In the fourth part, the parameter estimation for partially observed linear and nonlinear stochastic systems are considered, the expression of parameter estimators are obtained by using filtering method, and the strong consistency of the parameter estimator and asymptotic normality of the error of estimation are discussed. To be specific, the research content is as follows:In Chapter 1, the research background, motivation and research problems are discussed, and the content and contribution of the thesis are introduced.In Chapter 2, the almost sure parameter estimation for a class of continuous-time ergodic diffusion processes is studied from continuous-time observations. A closed interval on which the likelihood function is continuous and does not attain the maximum at endpoints of this interval is found. The existence of the maximum likelihood estimator is proved and the strong consistency of the estimator and the asymptotic normality of the error of estimation are discussed.In Chapter 3, the parameter estimation in probability for a class of continuous-time ergodic diffusion processes is investigated from continuous-time observations. An element is found in the neighborhood of the true parameter and the likelihood function gets the same value at endpoints of one small interval which includes the element. Rolle’s theorem is used to prove the existence of the maximum likelihood estimator. The consistency in probability of the maximum likelihood estimator and the asymptotic normality of the error of estimation are analyzed.In Chapter 4, the parameter estimation in probability for a class of continuous-time diffusion processes is discussed based on discrete observations. The approximate expression of the likelihood function is given by using Riemann-It(?) sum to replace the Lebesgue and It(?) integrals in the likelihood function. The consistency in probability of the maximum likelihood estimator is proved under the condition that the limitation of the approximate likelihood function attains the unique maximum at the true parameter value and the asymptotic normality of the error of estimation are studied.In Chapter 5, the parameter estimation for CI and CIR model is investigated by using different discretization methods. For CI model, the joint conditional probability density function for the Euler discretized process is given by applying It(?) lemma, the explicit expressions of the maximum likelihood estimator and the error of estimation are obtained, and the strong consistency of the estimators and the asymptotic normality of the error of estimation for the diffusion parameter are analyzed, the hypothesis testing is made to verify the effectiveness of the estimation method. For CIR model, the explicit expressions of the parameter estimators are obtained indirect by constructing a new variable and the strong consistency of the estimators are discussed, the simulation results illustrate that the estimation method is effective and the estimation precision is high.In Chapter 6, the parameter estimation for two kinds of partially observed linear stochastic systems are studied. For the linear stochastic system with single parameter, Kalman linear filtering and innovation theorem are used to obtain the explicit expressions of the state estimator and likelihood function, the explicit expression of the maximum likelihood estimator is provided, and the strong consistency of the parameter estimator and the asymptotic normality of the error of estimation are discussed. For the linear stochastic system with two parameters, the general filtering theory is applied to get the explicit expressions of the state estimator and likelihood function, the explicit expression of the parameter estimator is obtained and the strong consistency of the parameter estimator is analyzed.In Chapter 7, the problem of parameter estimation for partially observed nonlinear stochastic systems is considered. The suboptimal state estimation is obtained by constructing the extended Kalman filtering equation, the likelihood function is given based on the state estimation equation and the strong consistency of the parameter estimator is proved under the condition that the limitation of the likelihood function attains the unique maximum at the true parameter value.In Chapter 8, the results of the thesis are summarized and the potential research problems are put forward.