The Study Of Several Classes Of Markov Skeleton Processes And Some Properties For Q-processes

Markov processes is one class of important stochastic processes. Markov chain which is proposed by Russian mathematician A.A. Markov in 1907 is its primitive model.The core of Markov processes is Markov property which can be described as: known current state condition,the change of Markov processes in the future will not depend on previous change.Countable Markov processes is an active branch of Markov processes, and its research fruit is very abundant.The study of integral type random functional, numerical characteristic and Q-matrix is countable Markov processes important research contents.The Markov skeleton processes is stochastic processes which has Markov property at stopping time. Its area is more extensive than Markov processes and semi Markov processes and provide random models to more extensive applied problem. The Markov skeleton processes is proposed by professor Hou Zhengting etc in 1997.He and his students get fruitful works in this field.A series of articles and bibliographies have been published,for example bibliography [6],etc.The study of this thesis includes two aspects.One is to study several classes of Markov skeleton processes which include the semi Markov processes,the birth and death semi Markov processes,the birth and death type semi Markov skeleton processes, the other is to discuss important theories of the Q-processes and to construct one class of all-stable Q-processes.It consists of seven chapters.We drew the following conclusions.1. We study the one-dimensional distribution,integral type random functional and construction of the semi Markov processes.2. We introduce the concept of the birth and death semi Markov processes,and its numerals characteristics,and discuss up and down integral random functionals meanwhile.3. We introduce the definition of the birth and death type semi Markov skeleton processes,and get the initial distribution and the life distribution of Xⁿ(t,w)which imbed between skeleton sequence time rn_i(u;) and rn(w).We get the one-dimensional distribution of the birth and death type semi Markov skeleton processes. We make use of the one-dimensional distribution and initials distribution to construct the birth and death type semi Markov skeleton processes. We introduce the numerals characteristics of the birth and death type semi Markov skeleton processes ,and discuss their probability meanings.In finally,we study the up and down integral type random functionals.4. We study the decomposition of Markov processes and give the definition of Pooj(t).The necessary and sufficient condition of the Q-processes's B-condition is given.We introduce in Q-processes's figure characteristics ,and discuss probability meanings of the figure characteristics .We study the integral random functional of the Q-processes.By introducing the concept of the minimal processes,we have obtained two analytic construction theorems.5. We introduce the backward first arrival time and the forward first arrival time,and discuss their probability distributions and properties.Finally,we get the decomposition theorems of the forward taboo probability and the backward taboo probability.6. Q-matrix problem is discussed.We obtain equivalent conditions of Q-processes structure,and construct all-stable Q-processes and all-stable honest Q-processes about the same flow into on A(i).We get honest Q-processes's criterion of the existence.Finally,we get the analytical expression of (Q, II) processes.