| Processes of Ornstein-Uhlenbeck type are an important special case of Markov processes with jumps. Recently, they are widely used in describing the connections between branching processes and Levy processes, the stochastic volatilities of finance assets, and the default intensities and so on.The transition laws and the simulation algorithms of processes of Ornstein-Uhlenbeck type, and the parametric inference based on these discretely sampled processes are mainly studied in this dissertation. In detail, the main performances and results of this study are described as follows.(1) The definition of processes of Ornstein-Uhlenbeck type, the connections between processes of Ornstein-Uhlenbeck type and self-decomposable distributions, and the two kinds of methods to model processes of Ornstein-Uhlenbeck type are introduced. The modeling of these kinds of processes includes modeling via stationary distributions and modeling via background driving Levy processes. For modeling via background driving Levy processes, OU-CP processes, as a new category of processes of Ornstein-Uhlenbeck type, are provided along with some sub-categories.(2) The transition laws of processes of Ornstein-Uhlenbeck type are discussed. The characteristic function of the transition function is expressed. The transition laws of OU-CP processes and TS-OU processes are represented to be sum of the random variables, which have known distributions or closed-form densities. The forms of transition function and transition density of OU-CP processes and TS-OU processes are gotten with their smoothness. An explicit sequence of approximations to the transition density of OU-CP processes and TS-OU processes is also provided. Meanwhile, the self-decomposability of TS-OU processes is achieved.(3) The stochastic simulation algorithms and their realization methods of several processes of Ornstein-Uhlenbeck type are provided. Euler simulation scheme of processes of Ornstein-Uhlenbeck type, and the exact simulation of OU-CP processes with its realization method are introduced. The approximate simulation of TS-OU processes is improved. The exact simulation algorithm and its realization method of IG-OU processes are provided. An estimator of the intensity parameter of IG-OU processes is put forward, and its superiority to another estimator is evidenced by the exact simulations.(4) The likelihood method and the asymptotic properties of the estimator for parametric estimation of discretely sampled Gamma-OU processes are studied. An estimator of the intensity parameter is given, which convergence is stronger than weak convergence. Given the intensity parameter is estimated, a maximum likelihood estimator of shape parameter and scale parameter of the stationary distribution, and its existence, uniqueness, strong consistence and asymptotic normality, are studied. When estimating parameter with the maximum likelihood method, the likelihood function is not explicitly computable. By means of Gaver-Stehfest algorithm, an explicit sequence of approximations to the likelihood function is constructed, and that it converges at the true (but unknown) one is shown. Maximizing the sequence results in an estimator that converges at the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator.(5) The parametric estimations, which are by means of estimating functions, of several processes of Ornstein-Uhlenbeck type are studied. For OU-CP processes, after finding out some moment relations in describing the transition properties, the parameter of stationary distribution is estimated by the method of moments, and a consistent and asymptotically normal estimator is provided. Under some extra conditions, the estimation results can be generalized to the case of superposition. For IG-OU processes, after getting the conditional expectation and the conditional variance, the methodology of estimating the parameter of the processes, which is based on both simple estimating function and martingale estimating function, is given. The closed-form, the consistency and the asymptotic normality of the estimator are also provided. All the estimators in this paper are evidenced by simulations.The innovations of the achievements in this study are described as follows. To begin with, the systematic and concrete conclusions about the transition laws of several processes of Ornstein-Uhlenbeck type are gotten. Second, the exact simulation algorithm of IG-OU processes is provided. Third, by constructing an explicit sequence of approximations to the transition density of Gamma-OU processes, the approximate maximum likelihood estimation is found out. Finally, OU-CP processes, as a new category of processes of Ornstein-Uhlenbeck type, are put forward. Taking advantage of the moment connections of the processes between stationary distributions and each jump distributions in the background driving compound Poisson processes, the simple estimating function to estimate the parameter is constructed.The innovations of the methodologies in this study are described as follows. First, when studying the transition laws of processes of Ornstein-Uhlenbeck type, the idea to deal with the characteristic function of transition distribution function makes a good achievement. And with that, a random variable with unknown distribution is represented to be sum of the random variables, which have known distributions or closed-form densities. Second, as putting forward the likelihood function of the parameter of Gamma-OU processes, the method, which can be used to dispose of the likelihood functions with one or finite points of discontinuity in the population distribution functions, is provided. Third, Starting from researching the size connections between the density function and its derivatives with respect to the parameter, the existence of some moments related to the asymptotic variance of maximum likelihood estimator is proved. Finally, when building the martingale estimating function part of parametric estimation for discretely sampled IG-OU processes, the choice bypasses the commonly optimal linear estimating function and makes use of a property of processes of Ornstein-Uhlenbeck type.The conclusions of transition distribution laws of processes of Ornstein-Uhlenbeck type are helpful to the further cognitions of these processes for the researchers. Furthermore, those are benefit to the more extensive applications of the processes. For the simulation results of processes of Ornstein-Uhlenbeck type, it is an effectively experiential tool to evidence the conclusions of statistical inference of the processes. Undoubtedly, the maximum likelihood estimation of parameter in processes of Ornstein-Uhlenbeck type is a meaningful problem. That is not only because the maximum likelihood estimator generally has good properties, but also the maximum likelihood method can avoid the subjective choice of prior distributions in Bayesian method and the lack of precision in nonparametric method when the parametric model is known in advance. As a substitute of maximum likelihood estimation, the estimating function methods in parametric estimation of processes of Ornstein-Uhlenbeck type avoid the choice of optimal method, and reduce the time-consuming. When maximum likelihood estimation is difficult to get, the estimating function methods are better choices. In conclusion, the related studies of processes of Ornstein-Uhlenbeck type in this paper can not only perfect the statistical inference of the processes, but also enrich the effective methods to disposal of the real data.
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