Birth And Death Processes With Barriers And Compound Binomial Risk Models In Random Environment

This article is divided into two parts. The first part is about the research of the birth and death processes with barriers. Birth and death process is an ancient and classical random process. People have paid chronically high attention to the birth and death process, not only because the birth and death process has its own theo-retical significance and application value, but also it is the source of the generation of the thought and method of the general process. The second part introduces the thought of random environment, sets up a series of compound binomial risk mod-els in Markov-chain environment, mainly studies the ultimate ruin probability and the finite time ruin probability of the models.The first chapter is the introduction, it introduces the research background of this thesis, writing structure and innovative point and so on.The second chapter of this paper researches the qualitative theory of the birth and death processes(BDP) with barriers. In accordance with the classification of the barriers "0", classification of boundary point "Z", whether BDP are honest? whether BDP meet the backward or forward equations? There are many types of BDP. For each type of BDP, or has no, or has only one, or has an infinite number of BDP, and is the non-countable infinite number. Detail table is given.The third chapter gives the construction of the compound binomial risk mod-el in Markov-chain environment, which is abbreviated as MECM. The defini-tion of MECM in the Cossette(2004) is somewhat ambiguous, this paper points out it with a counterexample. On this basis, this chapter has improved and es-tablished strictly the compound binomial risk model in Markov-chain environ-ment (MECM)(0,I,B), and gives its characteristic4-tuple (ξ, τ(?),αI, FB). The new model here is more extensive than the model in the Cossette(2004). On the contrary, given one4-tuple (ξ, τ,α, F), this chapter proves that there exists MECM((?),I,B), and its characteristic4-tuple is just the above given (ξ,τ,α, F). Existence proof is constructive. Under the framework of the new model, we ob-tain the recursive formula for the finite time conditional non-ruin probability and the recursive formula for the conditional probability function of claim amount.The fourth chapter mainly considers a stable rate of return, namely R is con- stant, and positive, we establish the compound binomial risk model with constant interest in Markov-chain environment, and discuss claims, surplus, ruin probabil-ity of insurance company and so on, get some meaningful conclusions.The fifth chapter discusses different return rates or interest rates. We assume that the interest rate or rate of return is random, and is a Markov-chain, consider that the risk process of insurance company is a compound binomial model,and establish compound binomial risk models in Markov-chain interest rate environ-ment, and discuss the claims, surplus, ruin probability of insurance company and other issues, and obtain the recursion equations of finite time, infinite time condi-tional ruin probability.The sixth chapter assumes premium income is random, and is a compound binomial process, the influencing factor of premium income process, claim count-ing process and claim amount process is different,whether these processes are in-fluenced by environment and the depth of influence is also different. Therefore we assume that premium income is not affected by the environment, and claim count-ing process and claim amount process are affected by the environment, and these three processes do not affect each other. We establish the compound binomial risk model with random income in Markov-chain environment.In the framework of the model, we demonstrate the model’s existence, the process of the proof is constructive; discuss the claim, surplus, ruin probability in the risk model of insur-ance company, etc., and obtain recursive equations of the finite time, the infinite time conditional ruin probability.The seventh chapter have extended the discrete time risk model with the de-layed claims and threshold dividend. On the basis of Xie and Zou(2008), the fixed premium income is being promoted as a random premium, and we assume that the discount factor (or interest rate) of the bonus amount of each phase is a time ho-mogeneous Markov-chain with finite states, and research the expected discounted dividend amount under this risk model, and get an exact expression.