受Markov链调控的风险模型研究
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摘要
本论文建立并研究了几类受Markov链调控的风险模型.首先,从严格数学意义上证明了这些模型的存在性、概率构造和轨道刻划等.然后,在建立的这些数学模型的基础上,研究模型的破产问题,具体研究了各种破产量,特别是Gerber-Shiu折罚函数.
受Markov链调控的风险模型是经典风险模型的推广.经典风险模型只涉及两个随机过程(Poisson赔付计数过程和独立同分布赔付额过程),而且两个过程是独立的.本论文研究的受Markov链调控的风险模型,除了涉及两个随机过程(赔付计数过程和赔付额过程)外,还增加了一个Markov链(调控过程),甚至还有这三个过程以外的其它过程.这些过程不独立,而是相依的,它们之间满足某些相依关系.从数学角度看,风险模型的存在性问题是首先要解决的.只有证明了模型的存在性,才能在此基础上再研究模型的破产问题、最优控制问题等等.利用独立乘积空间技巧,经典风险模型的存在性是无容置疑的,数学上容易严格证明.但如果一个风险模型涉及两个或两个以上的互相依赖的随机过程(如受Markov链调控的风险模型),证明风险模型的存在性,就不能简单地用独立乘积空间技巧了.许多学者往往忽略这个问题,引进一些模型后就默认了模型的存在性.这是不可取的,也是不科学的.因此,本论文对新建立的受Markov链调控的风险模型,严格地证明了模型的存在性,并进一步研究了风险模型的概率构造及性质.从创新点和内容结构两个方面介绍如下.
一.论文的创新点.
本文的内容,除了第1章绪论和第2章2.1至2.4节外,文中所有其他的内容,即从第2章的2.5节直至本文最后的第7章,都是新的,都是作者的研究结果.具体地说,有下面7个方面的创新点.
1.提出用独立乘积空间构造相依随机变量的组装法
这里提出的“组装法”,自认为并不是大的发明.但本文却是首次系统地、明确地作为一种方法提出来,而且这个组装法在本文中得到很好的、很充分的应用,相信在构造其他的相依随机变量的情形也将是很有用的.
2.获得了Markov链的若干新结果
Markov链的理论研究已经很深入和丰富.虽然本论文主要是研究受Markov链调控的风险模型,但我们也获得了Markov链的若干新结果.
(1).研究了Markov链的一种随机时间替换.在一定假设下,时间替换后的链仍然具有Markov性.
一般情形Markov过程的时间替换早已有研究,要用到Markov过程的可加泛函等较高等的概念.本文研究的时间替换是在Markov链的初等框架下进行的.
(2).证明了：由q过程和Markov风险模型导出的几个多维过程是时间齐次Markov链；获得了一类有报酬的随机过程的Markov性和时间齐次性.
3.对Markov风险模型,获得了其Gerber-Shiu折罚函数满足的方程、递推公式和解析表达式.
4.首次引进双Markov风险模型,求出了它的生存概率和条件生存概率.
5.解决了Markov调制风险模型的存在性问题,给出了轨道刻划和概率构造.对附带税率的Markov调制过程,给出了轨道刻划.
6.给出了Markov相依风险模型的判别准则、必要条件和概率构造,解决了模型的存在性问题和判别问题.
7.新引进了半Markov相依风险模型,体现出“半”的意义.给出了模型的判别准则和一些必要条件,给出了概率构造,也解决了模型的存在性问题和判别问题.
二.论文的内容结构.
本论文总共7章,分两部分.
第一部分(第1-2章).首先,介绍本文必需要的关于Markov链的基本理论知识.然后,给出了作者获得的关于Markov链的若干新结果：关于q过程的一些新结果；Markov链的一种随机时间替换；有报酬的随机过程的Markov性和时间齐次性.再次,给出了用独立乘积空间构造相依随机变量的组装法.
第二部分(第3-7章).研究5类受Markov链调控的风险模型.每章研究一个模型,对于每个模型,研究的内容也不一样,各有侧重.
1. Markov风险模型(第3章)：研究它的Gerber-Shiu折罚函数,得到了Gerber-Shiu折罚函数满足的积分方程、递推公式、解析表达式.
2.双Markov风险模型(第4章)：研究它的生存概率和破产概率.首次提出双Markov风险模型.它是在Markov风险模型的基础上,将赔付额过程推广为一个Markov链.由于赔付时刻是一个Markov链(q过程)的跳跃点,赔付额也是一个Markov链,故提出的模型叫做双Markov风险模型.对于此模型的生存概率和条件生存概率,得到了它满足的积分方程、它的递推公式以及解析表达式.
3. Markov调制风险模型(第5章)：研究它的存在性和轨道结构.Markov调制风险模型前人已引进,但许多文献中,对于Markov环境的依赖都是描述性的、不很清晰.本文给出了Markov调制风险模型U=(A; J,S,X)的严格的数学定义,其中A=(C,Q,G,F)是模型U的特征组；给出了保费率、赔付时和赔付额依赖Markov环境的准确数学描述；模型的存在性和概率构造；对附带税率的Markov调制过程,给出了轨道刻划.
4. Markov相依风险模型(第6章)：研究它的判别准则、必要条件和概率构造,解决了模型的存在性和判别问题.Markov相依风险模型是前人提出的,它涉及3个相依的随机过程,而且3个随机过程的相依关系是用一个公式“一揽子”描述的.这样的3个过程,或者说,这样的模型是否存在?两两的关系如何?如何判断3个过程可否组成Markov相依风险模型?有无判别准则?本章给出了回答.
5.半Markov相依风险模型(第7章)：研究它的判别准则、必要条件和概率构造,解决了模型的存在性和判别问题.Albrecher and Boxma(2005)中已经引进了"Markov相依风险模型”,其文中虽然提到“半Markov性”,但没有体现出“半”的意义.本文引进的“半Markov相依风险模型”,体现了“半”的意义.而且,作为特殊情形包含了Markov相依风险模型,即本论文第6章中研究的模型.
Several risk models modulated by Markov chain are built and examined in this dissertation. Firstly, existence of these models, probabilistic construction and path-depict are proved in the strictly mathematic sense. Secondly, ruin problems of model, and ruin quantities, especially Gerber-Shiu discounted penalty function are probed.
The risk model modulated by Markov chain is a generalization of classical risk model. Classical risk model involves only two stochastic processes, which are Poisson counting process of claims and independent identically distributed claim amount process. Meanwhile, these two processes are independent. The risk model modulated by Markov chain, proposed in this dissertation, adds a Markov chain which is a new control process other than two stochastic processes mentioned above. Furthermore, other processes other than these three processes are also included. These processes are not independent but dependent. From the respective of math, the existence of risk models is desired to solve at first. The ruin problems of model, optimal control problems and so on can be solved only after the existence of the model is proved. The existence of classical risk model is undoubted which can be proved in strictly mathematic sense making use of independent product space technique. However, if a risk model involves two or more than two dependent stochastic processes, for example, risk model modulated by Markov chain, simply using independent product space technique cannot prove the existence of risk models. Therefore, many experts tend to neglect this issue and they take it for granted that the existence of model is default which is not scientific and not desirable as well. By contrast, this dissertation proves the existence of newly built risk model modulated by Markov chain at first, and then investigates the probabilistic construction and nature.
The innovation points and structure of this dissertation are as follows. I. Innovation Points
The main findings are originated by the author except Chapter1Introduction, Section2.1and2.4of Chapter2. There are seven innovation points in total.
1. Assembly method of constructing dependent random variables by using independent product space is put forward.
The assembly method mentioned here is not a big invention but is clarified as a method systematically at the first time in this dissertation. Furthermore, this method is fully applied in this dissertation and is hoped to be very useful in constructing other dependent random variables.
2. Several new findings of Markov chain are obtained.
Theory about Markov chain has been studied deeply and widely. This dissertation mainly attempts to explore the risk model modulated by Markov chain and meanwhile gets many new findings about Markov chain.
(1) A sort of replacement of random time for a Markov chain is investigated. Under a certain hypothesis, chain after replacement of time still has Markov property.
There have been many studies about time replacement of Markov process, using some advanced concept such as additive function of the Markov Process. The random time replacement studied in this dissertation is under the elementary framework of Markov chain.
(2) The proposition about multidimensional process derived by q process and Markov risk model is time-homogeneous Markov chain is proved. Moreover, Markov property and time-homogeneous property of stochastic process with reward are obtained.
3. For Markov risk model, the integral equation, recursive formulas and analytic expression of Gerber-Shiu discounted penalty function are derived.
4. Double-Markov risk model is introduced for the first time and survival probability and condition survival probability of it are solved.
5. The existence of Markov-modulated risk model is solved. Furthermore, Path-depict and probabilistic construction are defined. The Markov-modulated risk process with tax is given by Path-depict.
6. Criterion, necessary conditions and probabilistic construction of the Markov dependent risk model are proposed, which can settle the problems of model existence and model checking.
7. The newly introduced semi-Markov dependent risk model reflects the meaning of Semi. A criterion, necessary conditions of model and probabilistic construction are put forward to settle the problems of model existence and model checking. II Structure and Contents
This dissertation has seven chapters and is divided by two parts.
Part I (Chapter1and Chapter2). At first, the basic theory of Markov chain is introduced. Then, several new findings raised by the author are revealed:new results of q process, a sort of replacement of random time for a Markov chain and Markov property and time-homogeneous property of stochastic process with reward. At last, assembly method of constructing dependent random variables by using independent product space is put forward.
Part II (Chapter3-7). Five Markov-chain modulated risk models are explored. Each model is studied in each chapter respectively, with different emphasis.
1. Markov risk model (Chapter3):The integral equation, recursive formulas and analytic expression of Gerber-Shiu discounted penalty function are derived.
2. Double-Markov risk model (Chapter4):survival probability and ruin probability of it are explored. This model is extending the claim paid process to a Markov chain. Because claim paid moment is a jump point of Markov chain and claim paid amount is a Markov chain as well, the model is called double Markov risk model. The integral equation, recursive formulas and analytic expression of survival probability, condition survival probability are derived.
3. Markov-modulated risk model (Chapter5):The existence and Path-depict of it are explored. Though this model is introduced by predecessors, the Markov environment cannot stated more clearly in many papers. The strictly mathematic definitions of Markov-modulated risk model U=(A; J,S,X)are given, A=(C,Q,G,F) is characteristic group of model U among these. The premium, accurate mathematic description for dependent Markov environment for claim paid and claim amount, model existence and probabilistic construction are stated. The Path-depict is given for the Markov-modulated risk process with tax.
4. Markov dependent risk model (Chapter6):Criterion, necessary conditions and probabilistic construction of it are proposed, which can settle the problems of model existence and model checking. Markov dependent risk model is proposed by predecessors which involves three dependent Stochastic Processes. Meanwhile, the dependent relationship of these three stochastic processes is depicted by a package relation. This chapter answers several questions such as the existence of three processes, the relations of every two processes, how to judge whether the three processes can make up a Markov dependent risk model, is there any criterion to judge and so on.
5. Semi-Markov dependent risk model (Chapter7):Criterion, necessary conditions and probabilistic construction of it are proposed, which can settle the problems of model existence and model checking. Albrecher and Boxma (2005) have introduced Markov dependent risk model and mentioned Semi-Markov, but the meaning of Semi isn't reflected. The newly introduced semi-Markov dependent risk model, as the special case of Markov dependent risk model studied in Chapter6of this dissertation, reflects the meaning of Semi.
受Markov链调控的风险模型是经典风险模型的推广.经典风险模型只涉及两个随机过程(Poisson赔付计数过程和独立同分布赔付额过程),而且两个过程是独立的.本论文研究的受Markov链调控的风险模型,除了涉及两个随机过程(赔付计数过程和赔付额过程)外,还增加了一个Markov链(调控过程),甚至还有这三个过程以外的其它过程.这些过程不独立,而是相依的,它们之间满足某些相依关系.从数学角度看,风险模型的存在性问题是首先要解决的.只有证明了模型的存在性,才能在此基础上再研究模型的破产问题、最优控制问题等等.利用独立乘积空间技巧,经典风险模型的存在性是无容置疑的,数学上容易严格证明.但如果一个风险模型涉及两个或两个以上的互相依赖的随机过程(如受Markov链调控的风险模型),证明风险模型的存在性,就不能简单地用独立乘积空间技巧了.许多学者往往忽略这个问题,引进一些模型后就默认了模型的存在性.这是不可取的,也是不科学的.因此,本论文对新建立的受Markov链调控的风险模型,严格地证明了模型的存在性,并进一步研究了风险模型的概率构造及性质.从创新点和内容结构两个方面介绍如下.
一.论文的创新点.
本文的内容,除了第1章绪论和第2章2.1至2.4节外,文中所有其他的内容,即从第2章的2.5节直至本文最后的第7章,都是新的,都是作者的研究结果.具体地说,有下面7个方面的创新点.
1.提出用独立乘积空间构造相依随机变量的组装法
这里提出的“组装法”,自认为并不是大的发明.但本文却是首次系统地、明确地作为一种方法提出来,而且这个组装法在本文中得到很好的、很充分的应用,相信在构造其他的相依随机变量的情形也将是很有用的.
2.获得了Markov链的若干新结果
Markov链的理论研究已经很深入和丰富.虽然本论文主要是研究受Markov链调控的风险模型,但我们也获得了Markov链的若干新结果.
(1).研究了Markov链的一种随机时间替换.在一定假设下,时间替换后的链仍然具有Markov性.
一般情形Markov过程的时间替换早已有研究,要用到Markov过程的可加泛函等较高等的概念.本文研究的时间替换是在Markov链的初等框架下进行的.
(2).证明了：由q过程和Markov风险模型导出的几个多维过程是时间齐次Markov链；获得了一类有报酬的随机过程的Markov性和时间齐次性.
3.对Markov风险模型,获得了其Gerber-Shiu折罚函数满足的方程、递推公式和解析表达式.
4.首次引进双Markov风险模型,求出了它的生存概率和条件生存概率.
5.解决了Markov调制风险模型的存在性问题,给出了轨道刻划和概率构造.对附带税率的Markov调制过程,给出了轨道刻划.
6.给出了Markov相依风险模型的判别准则、必要条件和概率构造,解决了模型的存在性问题和判别问题.
7.新引进了半Markov相依风险模型,体现出“半”的意义.给出了模型的判别准则和一些必要条件,给出了概率构造,也解决了模型的存在性问题和判别问题.
二.论文的内容结构.
本论文总共7章,分两部分.
第一部分(第1-2章).首先,介绍本文必需要的关于Markov链的基本理论知识.然后,给出了作者获得的关于Markov链的若干新结果：关于q过程的一些新结果；Markov链的一种随机时间替换；有报酬的随机过程的Markov性和时间齐次性.再次,给出了用独立乘积空间构造相依随机变量的组装法.
第二部分(第3-7章).研究5类受Markov链调控的风险模型.每章研究一个模型,对于每个模型,研究的内容也不一样,各有侧重.
1. Markov风险模型(第3章)：研究它的Gerber-Shiu折罚函数,得到了Gerber-Shiu折罚函数满足的积分方程、递推公式、解析表达式.
2.双Markov风险模型(第4章)：研究它的生存概率和破产概率.首次提出双Markov风险模型.它是在Markov风险模型的基础上,将赔付额过程推广为一个Markov链.由于赔付时刻是一个Markov链(q过程)的跳跃点,赔付额也是一个Markov链,故提出的模型叫做双Markov风险模型.对于此模型的生存概率和条件生存概率,得到了它满足的积分方程、它的递推公式以及解析表达式.
3. Markov调制风险模型(第5章)：研究它的存在性和轨道结构.Markov调制风险模型前人已引进,但许多文献中,对于Markov环境的依赖都是描述性的、不很清晰.本文给出了Markov调制风险模型U=(A; J,S,X)的严格的数学定义,其中A=(C,Q,G,F)是模型U的特征组；给出了保费率、赔付时和赔付额依赖Markov环境的准确数学描述；模型的存在性和概率构造；对附带税率的Markov调制过程,给出了轨道刻划.
4. Markov相依风险模型(第6章)：研究它的判别准则、必要条件和概率构造,解决了模型的存在性和判别问题.Markov相依风险模型是前人提出的,它涉及3个相依的随机过程,而且3个随机过程的相依关系是用一个公式“一揽子”描述的.这样的3个过程,或者说,这样的模型是否存在?两两的关系如何?如何判断3个过程可否组成Markov相依风险模型?有无判别准则?本章给出了回答.
5.半Markov相依风险模型(第7章)：研究它的判别准则、必要条件和概率构造,解决了模型的存在性和判别问题.Albrecher and Boxma(2005)中已经引进了"Markov相依风险模型”,其文中虽然提到“半Markov性”,但没有体现出“半”的意义.本文引进的“半Markov相依风险模型”,体现了“半”的意义.而且,作为特殊情形包含了Markov相依风险模型,即本论文第6章中研究的模型.
Several risk models modulated by Markov chain are built and examined in this dissertation. Firstly, existence of these models, probabilistic construction and path-depict are proved in the strictly mathematic sense. Secondly, ruin problems of model, and ruin quantities, especially Gerber-Shiu discounted penalty function are probed.
The risk model modulated by Markov chain is a generalization of classical risk model. Classical risk model involves only two stochastic processes, which are Poisson counting process of claims and independent identically distributed claim amount process. Meanwhile, these two processes are independent. The risk model modulated by Markov chain, proposed in this dissertation, adds a Markov chain which is a new control process other than two stochastic processes mentioned above. Furthermore, other processes other than these three processes are also included. These processes are not independent but dependent. From the respective of math, the existence of risk models is desired to solve at first. The ruin problems of model, optimal control problems and so on can be solved only after the existence of the model is proved. The existence of classical risk model is undoubted which can be proved in strictly mathematic sense making use of independent product space technique. However, if a risk model involves two or more than two dependent stochastic processes, for example, risk model modulated by Markov chain, simply using independent product space technique cannot prove the existence of risk models. Therefore, many experts tend to neglect this issue and they take it for granted that the existence of model is default which is not scientific and not desirable as well. By contrast, this dissertation proves the existence of newly built risk model modulated by Markov chain at first, and then investigates the probabilistic construction and nature.
The innovation points and structure of this dissertation are as follows. I. Innovation Points
The main findings are originated by the author except Chapter1Introduction, Section2.1and2.4of Chapter2. There are seven innovation points in total.
1. Assembly method of constructing dependent random variables by using independent product space is put forward.
The assembly method mentioned here is not a big invention but is clarified as a method systematically at the first time in this dissertation. Furthermore, this method is fully applied in this dissertation and is hoped to be very useful in constructing other dependent random variables.
2. Several new findings of Markov chain are obtained.
Theory about Markov chain has been studied deeply and widely. This dissertation mainly attempts to explore the risk model modulated by Markov chain and meanwhile gets many new findings about Markov chain.
(1) A sort of replacement of random time for a Markov chain is investigated. Under a certain hypothesis, chain after replacement of time still has Markov property.
There have been many studies about time replacement of Markov process, using some advanced concept such as additive function of the Markov Process. The random time replacement studied in this dissertation is under the elementary framework of Markov chain.
(2) The proposition about multidimensional process derived by q process and Markov risk model is time-homogeneous Markov chain is proved. Moreover, Markov property and time-homogeneous property of stochastic process with reward are obtained.
3. For Markov risk model, the integral equation, recursive formulas and analytic expression of Gerber-Shiu discounted penalty function are derived.
4. Double-Markov risk model is introduced for the first time and survival probability and condition survival probability of it are solved.
5. The existence of Markov-modulated risk model is solved. Furthermore, Path-depict and probabilistic construction are defined. The Markov-modulated risk process with tax is given by Path-depict.
6. Criterion, necessary conditions and probabilistic construction of the Markov dependent risk model are proposed, which can settle the problems of model existence and model checking.
7. The newly introduced semi-Markov dependent risk model reflects the meaning of Semi. A criterion, necessary conditions of model and probabilistic construction are put forward to settle the problems of model existence and model checking. II Structure and Contents
This dissertation has seven chapters and is divided by two parts.
Part I (Chapter1and Chapter2). At first, the basic theory of Markov chain is introduced. Then, several new findings raised by the author are revealed:new results of q process, a sort of replacement of random time for a Markov chain and Markov property and time-homogeneous property of stochastic process with reward. At last, assembly method of constructing dependent random variables by using independent product space is put forward.
Part II (Chapter3-7). Five Markov-chain modulated risk models are explored. Each model is studied in each chapter respectively, with different emphasis.
1. Markov risk model (Chapter3):The integral equation, recursive formulas and analytic expression of Gerber-Shiu discounted penalty function are derived.
2. Double-Markov risk model (Chapter4):survival probability and ruin probability of it are explored. This model is extending the claim paid process to a Markov chain. Because claim paid moment is a jump point of Markov chain and claim paid amount is a Markov chain as well, the model is called double Markov risk model. The integral equation, recursive formulas and analytic expression of survival probability, condition survival probability are derived.
3. Markov-modulated risk model (Chapter5):The existence and Path-depict of it are explored. Though this model is introduced by predecessors, the Markov environment cannot stated more clearly in many papers. The strictly mathematic definitions of Markov-modulated risk model U=(A; J,S,X)are given, A=(C,Q,G,F) is characteristic group of model U among these. The premium, accurate mathematic description for dependent Markov environment for claim paid and claim amount, model existence and probabilistic construction are stated. The Path-depict is given for the Markov-modulated risk process with tax.
4. Markov dependent risk model (Chapter6):Criterion, necessary conditions and probabilistic construction of it are proposed, which can settle the problems of model existence and model checking. Markov dependent risk model is proposed by predecessors which involves three dependent Stochastic Processes. Meanwhile, the dependent relationship of these three stochastic processes is depicted by a package relation. This chapter answers several questions such as the existence of three processes, the relations of every two processes, how to judge whether the three processes can make up a Markov dependent risk model, is there any criterion to judge and so on.
5. Semi-Markov dependent risk model (Chapter7):Criterion, necessary conditions and probabilistic construction of it are proposed, which can settle the problems of model existence and model checking. Albrecher and Boxma (2005) have introduced Markov dependent risk model and mentioned Semi-Markov, but the meaning of Semi isn't reflected. The newly introduced semi-Markov dependent risk model, as the special case of Markov dependent risk model studied in Chapter6of this dissertation, reflects the meaning of Semi.
引文
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[5]Albrecher, H., Renaud, J. & X.Zhou. A Levy insurance risk process with tax [J]. Journal of Applied Probability,2008,45(2):363-375.
[6]Albrecher, H. & O.J. Boxma. On the discounted penalty function in a Markov-dependent risk model [J]. Insurance:Mathematics and Economics, 2005,37:650-672.
[7]Albrecher, H., Hartinger, J. & R. F. Ticky. On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier [J]. Scantinavian Acturial Journal, 2005,2:103-126.
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[9]Albrecher, H. & O. Boxma. A ruin model with dependence between claim sizes and cliam intervals [J]. Insurance:Mathematics and Economics, 2004,35 (2):245-254.
[10]Azcue, P. & N. Muler. Optional reinsurance and dividend distribution policies in the Cramer-Lundberg model [J]. Mathematical Finance, 2005,15 (2):261-368.
[11]Asmussen, S. Ruin probabilities [M], Singapore:World Scientific, 2000.
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[13]Asmussen, S. Risk theory in a Markovian environment [J]. Scandinavia Actuarial Journal,1989,2:69-100.
[14]Adan, I. & V. Kulkarni. Single-server queue with Markov dependent inter-arrival and service times [J]. Queuing Systems,2003,45: 113-114.
[15]Avram, F. & M. Usabel. Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times [J]. ASTIN Bulle tin, 2004,34 (2):315-332.
[16]Adan, I. & V. Kulkarni. Single-server queue with Markov dependent inter-arrival and service times [J]. Queuing Systems,2003,45: 113-114.
[17]Benouaret, Z. & D. Aissani. Strong stability in a two-dimensional classical risk model with independent claims [J]. Scandinavian Actuarial Journal,2010,2:83-92.
[18]Cheung, E. Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times [J]. Insurance:Mathematics and Economics,2013,53:343-354.
[19]Cheung, E. Analysis of a generalized penalty function in a semi-Markovian risk model [J]. North American Actuaries Journal,2009, 13(4):497-513.
[20]Cheung, E. Analysis of some risk models involving dependence. PHD Thesis, University of Waterloo,2010.
[21]Cramer, H. On the mathematical theory of risk [M]. In:Skandia jubilee volume, Stockholm,1930.
[22]Chung, K. L. Markov chains with stationary transition probabilities [M]. Berlin:Springer-Verlag,1967.
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[28]Doob, J. L. Stochastic Processes [M]. John Wiley sons,1953.
[29]Dwyer & F.Robert. Developing buyer seller relations[J]. Journal of Marketing,1987,51 (5):11-28.
[30]Elena, A. An application of semi-Markovian models to the ruin problem [J]. Revista colombiana de stadistica,2011,34(3)-.477-495.
[31]Frostig, E., Susan, M., Pitts, & K. Politis. The time to ruin and the number of claims until ruin for phase-type claims [J]. Insurance: Mathematics and Economics,2012,51 (1):19-25.
[32]Guillou, A., Loisel, S., & G. Stupf ler. Estimation of the parameters of a Markov-modulated loss process in insurance [J]. Insurance: Ma thema tics and Economics,2013,53 (2):388-404.
[33]Gerber, H. & E. Shiu. The time value of ruin in Sparre Andersen model [J]. North American Actuarial Journal,2005,9(2):49-69.
[34]Gerber, H. & E.Shiu. On the time value of ruin [J]. North American Actuarial Journal,1998,2(1):48-72.
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[45]Li, S. M. & L. Yi. On the generalized Gerber-Shiu function for surplus processes with interest [J]. Insurance:Mathematics and Economics, 2013,52(2):127-134.
[46]Lundberg, F. Uber die theorie derruckversicherung [J]. Trans. Ⅵ Int Conger Actuarial,1909,1:877-948.
[47]Li, S. & J. Garrido. On ruin for Erlang(n) risk model [J]. Insurance: Mathematics and Economics,2004,34(3):391-408.
[48]Li, S. & J. Garrido. On a general class of renewal risk process: analysis of the Gerber-Shui function[J]. Advances in Applied probability,2005,37(3):836-856.
[49]Lu, X. W. & W. Jiang. The model of Markov process for customer relations development and application [J]. Industrial Engineering and Management,2004,9 (1):40-44.(路晓伟,蒋馥.客户关系发展的马尔柯夫过程模型及其应用[J].工业工程与 管理,2004,9(1)：40-44.)
[50]Mo, X. Y. Assembly method constructing dependent random variables with independent product space [J]. Journal of Natural Science of Hunan Normal University,2010,33(2):3-6. (in chinese)(莫晓云.用独立乘积空间构造相依随机变量的组装法[J].湖南师范大学学报自然科学学报,2010,33(2)：3-6.)
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