The Statistical Problems On Some Markov Processes In Random Environment

A variate of phenomenon in a variate fields, such as biology, operation research, demography, economics, and engineering, can be well modeled by birth-death processes and branching processes. At the same time, the richness of the theory of birth-death process provides standard analytical methods to investigate numerous important quantities in general stochastic process and practice, such as stationary distribution, mean first passage time, etc. Motivated by this, a lot of mathematicians, biologists, engineers, etc., have made great efforts in it. Nevertheless, a lot of issues are still in the way being solved.This thesis consists of four parts. In part Ⅰ (Chapter 1-2), we focuss on investigating a special linear population process—immigration-birth-death process. First, in Chapter 1, we provide some exact, elegant formulae of the transition distribution of immigration-birth-death process by moment generating function. They are very useful in simulation and statistical inference. Second, in Chapter 2, by the exact descriptions we obtained, the maximum likelihood estimation under two sampling schemes: complete observation and equidistant observation, are given and in the end, simulations are carried out to test the bias and efficiency of our estimators.In part Ⅱ (Chapter 3-4), we deal with another special interesting nonlinear population process—prey-predator process. First, in Chapter 3, a two-dimensional interacting birth-death process is established and by means of two-dimensional moment generating function, the equivalence in mean between our model and the classic prey-predator model are proved. Also, in Chapter 4 chapter, using the properties of stochastic prey-predator model, the maximum likelihood inference under two sampling schemes are given and simulations are carried out.In part Ⅲ (Chapter 5 ), the theory of classic birth-death process is generalized to random environment. Our main result is for any immigration-emigration birth and death matrix random environment, q, with birth rateless than death rate, there are a unique q-process in random environment P(9*(0);t) and a bi-immigration birth and death process in random environment (X*,f) with random transition matrix P(9*{Q);t) such that P(9*(0);t) is ergodic and X* is a strictly stationary process.In part IV (Chapter 6-7), we discuss r-dimension branching chains in random environment. First, in Chapter 6, we define some probability generating functions in random environment and use them to give a exact formula of the the covariance matrix of r-dimension branching chains in random environment. Second, in Chapter 7, we define the Laplace functional of r-dimension branching chains in random environment and then discuss the moments of r-dimension branching chains in random environment.