带壁生灭过程及随机环境中复合二项风险模型
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摘要
本文写作共分为两大部分。第一部分是关于带壁生灭过程的研究。生灭过程是一个古老而经典的随机过程,人们长期地高度关注生灭过程,不仅因为生灭过程有它本身的理论意义和应用价值,而且它是产生研究一般过程的思想和方法的源泉。第二部分引入随机环境的思想,建立了一系列马氏链环境中的复合二项风险模型,主要研究了其最终破产概率和有限时间破产概率。
第一章为绪论,综合介绍了本论文的研究背景、论文写作结构、创新点等。
第二章研究了带壁生灭过程(BDP)的定性理论。按照壁0的分类,边界点z的分类,BDP是否诚实,是否满足向后或向前方程组,存在许多组合类型的BDP。对于每一种类型的BDP,或者没有,或者仅一个,或者有无穷多个BDP,而且是非可数的无穷多个。列出了一个详细的表。
第三章构建马氏链环境中复合二项风险模型,简记为MECM.文献Cossette(2004)中对MECM的定义有些含混,本章以反例指出这一点,在此基础上改进并严格建立了马氏链环境中复合二项风险模型MECM((?),I,B),给出其特征四元组(ξ,τ(?),αI,FB)本章的模型较文献Cossette(2004)广泛.反之,给定一个四元组(ξ,τ,a,F),本章证明了：存在MECM((?),I,B),其特征四元组与给定的(ξ,r,α,F)重合.存在性证明是构造性的.在新的模型框架下,得出了有限时间的条件非破产概率递推公式及赔付额的条件概率函数的递推公式.
第四章主要考虑有稳定的回报率情况,即R是常数,且为正的,我们建立了带常利率的马氏链环境复合二项风险模型,讨论保险公司的赔付、盈余、破产概率等问题,得到了一些有意义的结论。
第五章讨论回报率或利率不同的情况。我们假定利率或回报率是随机的,且是马氏链的情形,考虑保险公司风险过程为复合二项模型的情况,建立了马氏链利率环境复合二项风险模型,讨论保险公司的赔付、盈余、破产概率等问题,得到了有限时间、无限时间的条件破产概率的递推方程。
第六章假定保费收入是随机的,且是复合二项过程,保费收入过程与赔付计数过程、赔付额过程的影响因素不尽相同,受到环境的影响及程度亦是不同的,因此假定保费收入不受该环境的控制,而赔付过程、赔付额过程受到环境的控制,保费收入、赔付次数、赔付额互相不影响,我们建立了马氏链环境中带随机收入的复合二项风险模型,在模型框架下,证明了模型的存在性,证明过程是构造性的；讨论了保险公司风险模型的赔付、盈余、破产概率等问题,得到了有限时间、无限时间条件破产概率的递推方程。
第七章对具有延迟索赔和门槛分红的离散时间风险模型进行了推广,在Xie和Zou(2008)的基础上,将固定保费收入推广为随机保费的情况,并假设其中每个阶段的分红量的折现因子(或利率)是一个有限状态的时齐马氏链,我们研究了该风险模型下的期望折现分红总量,得到了它的一个精确表达式。
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.
第一章为绪论,综合介绍了本论文的研究背景、论文写作结构、创新点等。
第二章研究了带壁生灭过程(BDP)的定性理论。按照壁0的分类,边界点z的分类,BDP是否诚实,是否满足向后或向前方程组,存在许多组合类型的BDP。对于每一种类型的BDP,或者没有,或者仅一个,或者有无穷多个BDP,而且是非可数的无穷多个。列出了一个详细的表。
第三章构建马氏链环境中复合二项风险模型,简记为MECM.文献Cossette(2004)中对MECM的定义有些含混,本章以反例指出这一点,在此基础上改进并严格建立了马氏链环境中复合二项风险模型MECM((?),I,B),给出其特征四元组(ξ,τ(?),αI,FB)本章的模型较文献Cossette(2004)广泛.反之,给定一个四元组(ξ,τ,a,F),本章证明了：存在MECM((?),I,B),其特征四元组与给定的(ξ,r,α,F)重合.存在性证明是构造性的.在新的模型框架下,得出了有限时间的条件非破产概率递推公式及赔付额的条件概率函数的递推公式.
第四章主要考虑有稳定的回报率情况,即R是常数,且为正的,我们建立了带常利率的马氏链环境复合二项风险模型,讨论保险公司的赔付、盈余、破产概率等问题,得到了一些有意义的结论。
第五章讨论回报率或利率不同的情况。我们假定利率或回报率是随机的,且是马氏链的情形,考虑保险公司风险过程为复合二项模型的情况,建立了马氏链利率环境复合二项风险模型,讨论保险公司的赔付、盈余、破产概率等问题,得到了有限时间、无限时间的条件破产概率的递推方程。
第六章假定保费收入是随机的,且是复合二项过程,保费收入过程与赔付计数过程、赔付额过程的影响因素不尽相同,受到环境的影响及程度亦是不同的,因此假定保费收入不受该环境的控制,而赔付过程、赔付额过程受到环境的控制,保费收入、赔付次数、赔付额互相不影响,我们建立了马氏链环境中带随机收入的复合二项风险模型,在模型框架下,证明了模型的存在性,证明过程是构造性的；讨论了保险公司风险模型的赔付、盈余、破产概率等问题,得到了有限时间、无限时间条件破产概率的递推方程。
第七章对具有延迟索赔和门槛分红的离散时间风险模型进行了推广,在Xie和Zou(2008)的基础上,将固定保费收入推广为随机保费的情况,并假设其中每个阶段的分红量的折现因子(或利率)是一个有限状态的时齐马氏链,我们研究了该风险模型下的期望折现分红总量,得到了它的一个精确表达式。
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.
引文
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