On the Transition Law of O -- U Compound Poisson Processes

O-U compound Poisson processes, as a new category of processes of Ornstein--Uhlenbeck type, are put forward in this paper. These processes are a generalization of gamma O--U processes. By dealing with the characteristic function of the transition distribution function, the transition law of the O--U compound Poisson process is expressed by a sum of Poisson-distributed number of random variables, which are independent identically-distributed and have the common known density function. And we also obtain the expressions of both the transition function and the transition density of the process. The conclusions in this paper are the theoretical foundations for further researches on statistical inference of the process.