保费随机收入的二维风险模型的破产问题
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摘要
风险理论是保险数学中的重要理论,而破产论又是风险理论的核心.破产论最早提出的风险模型是Lundberg-Cramer经典破产模型,随着人们对这一模型全面而系统的研究,使经典论的研究内容不断得到深化.如,保费收入随机化的风险模型就是其中改进后得到的一种新模型.本文将保费随机收入的一维风险模型推广到二维风险模型,即为保费随机收入的二维风险模型.但是二维风险模型并不是一维风险模型的简单推广,它要比一维风险模型复杂得多.本文用两种方法探讨了保费随机收入的二维风险模型的破产问题.
根据内容本文分为以下三章：
第一章为绪论,主要介绍风险模型发展的历史、二维风险模型的研究现状和部分相关的概率知识.
第二章讨论了保费随机收入的二维风险模型的破产问题.首先用鞅方法求出它的Lundberg型上界然后假定当保险公司收取的两种保费(C1j,C2j),J=1,2,…和支付的两种索赔(X1j,X2j),j=1,2,…均服从指数分布,并且它们的联合分布函数F1(c1j,c2j),j=1,2,…和F2(x1j,x2j),j=1,2,…均属于二元的Farlie-Gumbel-Morgenstern类,当关联系数p1和p2发生变化时,通过数据分析了破产概率上界的变化情况.
第三章主要是用一维风险过程来讨论二维风险模型的破产问题.首先,定义一个一维风险模型其中参数(a1,a2)满足ai>0,i=1,2且∑i=12ai=1.这样,第二章中定义的二维风险模型转化成了一维模型的形式,其中从而,运用一维风险过程的结果得到了保费随机收入的二维风险模型最终破产概率满足的Lundberg不等式及最终破产概率满足的解析式然后,运用一维风险过程的方法推导出了它的Gerber-Shiu罚金折现函数Φ(ua)满足的积分方程并求出Φ(ua)在n=1时的Laplace变换之后验证了Φ(r)在特殊情况下与一维风险模型中的结果是一致的.最后探讨了保费随机收入的二维风险模型推广了的Gerber-Shiu罚金折现函数
Risk theory is an important part in insurance mathematics, and ruin theory is the main content in risk theory. Lundberg-Cramer classical risk model is the first-emerging model in ruin theory. The contents of classical theory are deepened with people's com-prehensive and systematic research. For example, the risk model with random premium income is one of improved models. In this paper, a one-dimensional risk model with random premium income is extended to a two-dimensional risk model which is a two-dimensional risk model with random premium income. Of course, the two-dimensional risk model is not a simple extension of the one-dimensional risk model and it is more complicated than a one-dimensional model. In this thesis, we discuss the ruin problem of the two-dimensional risk model with stochastic premium through two methods.
This thesis is divided into three chapters according to contents:
Chapter 1 is prolegomenon, in which we mainly introduce the developmental history of risk model, the current situation of a two-dimensional risk model's research and part of interrelated knowledge of probability.
In chapter 2, we discuss the ruin probability of a two-dimensional risk model with random premium income which is First, we obtain a Lundberg-type upper bound through the martingale technique. Then, by supposing that two kinds of premiums (C1j,C2j), j= 1,2,…and two kinds of claims (X1j,X2j), j= 1,2,…all obey ex-ponential distribution and their joint distribution functions F1(c1j,C2j),j= 1,2,…and F2(x1j,X2j),j= 1,2,…belong to bivariate Farlie-Gumbel-Morgenstern class, we analyse the upper bound's change of infinite-time ruin probabality by analysis of data when the correlation coefficientρ1 andρ2 change.
In chapter 3, we mainly consider the ruin problem of the two-dimensional risk model through methods of the one-dimensional risk process. Firstly, we define a one-dimensional risk model where ai> 0,i= 1,2, and∑i=1 2 ai= 1.So the two-dimensional risk model defined in Chapter 2 is changed into the one-dimensional form Then we get the Lundberg inequality and the infinite-time ruin probability using some results of the one-dimensional risk process. Secondly, we obtain the integral equation of the Gerber-Shiu discounted penalty function fC(c1)fC(c2)…fC(cn)dc1dc2…dcn and the Laplace transform ofΦ(ua) if n = 1, which is Next, we verify that the result ofΦ(r) is consistent with the one-dimensional risk model's in special cases. Finally, we discuss the extended Gerber-Shiu discounted penalty function of the two-dimensional risk model with random premium income
根据内容本文分为以下三章：
第一章为绪论,主要介绍风险模型发展的历史、二维风险模型的研究现状和部分相关的概率知识.
第二章讨论了保费随机收入的二维风险模型的破产问题.首先用鞅方法求出它的Lundberg型上界然后假定当保险公司收取的两种保费(C1j,C2j),J=1,2,…和支付的两种索赔(X1j,X2j),j=1,2,…均服从指数分布,并且它们的联合分布函数F1(c1j,c2j),j=1,2,…和F2(x1j,x2j),j=1,2,…均属于二元的Farlie-Gumbel-Morgenstern类,当关联系数p1和p2发生变化时,通过数据分析了破产概率上界的变化情况.
第三章主要是用一维风险过程来讨论二维风险模型的破产问题.首先,定义一个一维风险模型其中参数(a1,a2)满足ai>0,i=1,2且∑i=12ai=1.这样,第二章中定义的二维风险模型转化成了一维模型的形式,其中从而,运用一维风险过程的结果得到了保费随机收入的二维风险模型最终破产概率满足的Lundberg不等式及最终破产概率满足的解析式然后,运用一维风险过程的方法推导出了它的Gerber-Shiu罚金折现函数Φ(ua)满足的积分方程并求出Φ(ua)在n=1时的Laplace变换之后验证了Φ(r)在特殊情况下与一维风险模型中的结果是一致的.最后探讨了保费随机收入的二维风险模型推广了的Gerber-Shiu罚金折现函数
Risk theory is an important part in insurance mathematics, and ruin theory is the main content in risk theory. Lundberg-Cramer classical risk model is the first-emerging model in ruin theory. The contents of classical theory are deepened with people's com-prehensive and systematic research. For example, the risk model with random premium income is one of improved models. In this paper, a one-dimensional risk model with random premium income is extended to a two-dimensional risk model which is a two-dimensional risk model with random premium income. Of course, the two-dimensional risk model is not a simple extension of the one-dimensional risk model and it is more complicated than a one-dimensional model. In this thesis, we discuss the ruin problem of the two-dimensional risk model with stochastic premium through two methods.
This thesis is divided into three chapters according to contents:
Chapter 1 is prolegomenon, in which we mainly introduce the developmental history of risk model, the current situation of a two-dimensional risk model's research and part of interrelated knowledge of probability.
In chapter 2, we discuss the ruin probability of a two-dimensional risk model with random premium income which is First, we obtain a Lundberg-type upper bound through the martingale technique. Then, by supposing that two kinds of premiums (C1j,C2j), j= 1,2,…and two kinds of claims (X1j,X2j), j= 1,2,…all obey ex-ponential distribution and their joint distribution functions F1(c1j,C2j),j= 1,2,…and F2(x1j,X2j),j= 1,2,…belong to bivariate Farlie-Gumbel-Morgenstern class, we analyse the upper bound's change of infinite-time ruin probabality by analysis of data when the correlation coefficientρ1 andρ2 change.
In chapter 3, we mainly consider the ruin problem of the two-dimensional risk model through methods of the one-dimensional risk process. Firstly, we define a one-dimensional risk model where ai> 0,i= 1,2, and∑i=1 2 ai= 1.So the two-dimensional risk model defined in Chapter 2 is changed into the one-dimensional form Then we get the Lundberg inequality and the infinite-time ruin probability using some results of the one-dimensional risk process. Secondly, we obtain the integral equation of the Gerber-Shiu discounted penalty function fC(c1)fC(c2)…fC(cn)dc1dc2…dcn and the Laplace transform ofΦ(ua) if n = 1, which is Next, we verify that the result ofΦ(r) is consistent with the one-dimensional risk model's in special cases. Finally, we discuss the extended Gerber-Shiu discounted penalty function of the two-dimensional risk model with random premium income
引文
[1]Lundberg F. Approximerad framstallning av sannolikhetsfunktionen[C]. Aterforsakering av Kollektivrisker, Almqvist & Wiksell, Uppsala,1903.
[2]Cramer, H. On the mathematical theory of risk[M]. Stoockholm:Skandia Jubilee Volume, 1930.
[3]Cramer, H. Collective risk theory[M]. Stockholm:Skandia Jubilee Volume,1985.
[4]Gerber, H.U.著.成世学,严颖译.数学风险论导引[M].北京：世界图书出版公司,1997.
[5]成世学.破产论研究综述[J].数学进展,2002,31(5)：403-422.
[6]王广华,吕玉华.保费收入随机化的风险模型[J].经济数学,2006,23(3)：221-228.
[7]Chan, W.-S., Yang, H., Zhang, L. Some results on ruin probabilities in a two-dimensional risk model[J]. Insurance:Mathematics and Economics,2003,32:345-358.
[8]Dang, L., Zhu, N., Zhang, H. Survival probability for a two-dimensional risk model[J]. Insurance:Mathematics and Economics,2009,44(3):491-496.
[9]Li, J., Liu, Z., Tang, Q. On the ruin probabilities of a bidimensional perturbed risk model[J] Insurance:Mathematics and Economics,2007,41:185-195.
[10]Yuen, K.C., Guo, J.Y., Wu, X.Y. On the first time of ruin in the bivariate compound Poisson model[J] Insurance:Mathematics and Economics,2006,38(2):298-308.
[11]王泉,张奕.考虑常利率的二维离散风险模型的破产概率[J].浙江大学学报(理学版),2008,35(5)：501-506.
[12]候丽娟.一类二维风险模型的破产概率[J].泰山学院报,2008,30(6)：48-52.
[13]马学思,刘次华.双Poisson二维风险模型的破产概率[J].统计与决策,2006,(16)：24-25.
[14]李祥发,宋会杰,郭鹏江.扰动的二维风险模型的破产概率[J].西北大学学报(自然科学版),2008,38(6)：885-888.
[15]刘立国,王炳昌,龙国军.重尾索赔下二维风险模型有限时间的破产概率[J].数学理论与应用,2007,27(2)：67-70.
[16]马学思,刘次华.二维相关风险模型的破产概率[J].统计与决策,2009,(2)：151-152.
[17]张旭.具有两类相关索赔的二元风险模型的若干问题[J].重庆工学院报(自然科学版),2008,22(3)：148-152.
[18]Fima C Klebaner. Introduction to stochastic calculus with applications[M]. London:Im-perial College Press,1998.
[19]Nelsen, R.B. An introduction to copulas[M]. Second ed., In:Springer-Series in Statistics, Springer-Verlag, New York,2006.
[20]Kotz, S., Balakrishnan, N., Johnson, N.L. Continuous multivariate distribution, Vol.1. Models and applications[M]. Second ed., Wiley-Interscience, New York,2000.
[21]Cossette, H., Marceau, E., Marri, F. On the compound Poisson risk model with dependence. based on a generalized Farlie-Gumbel-Morgenstern copula[J]. Insurace:Mathematics and Economics,2008,43:444-455.
[22]Drouet mari, D., Kotz, S. Correlation and dependence[M]. Imperial College Press,2001.
[23]Gerber, H.U., Shiu, E.S.W. On the time value of ruin[J]. North American Acturial Journal, 1998,30:48-78.
[24]Willmot, G.E., Lin. Lundberg approximations for compound distributions with insurance application[M]. Springer-Verlag, New York,2001.
[25]Yuen, K.C., Guo, J.Y., Wu, X.Y. On a correlated aggregate claims model with Poisson and Erlang risk process[J]. Insurance:Mathematics and Economics,2002,31:205-214.
[26]Cossette, H., Marceau, E. The discrete-time risk model with correlate classes of business[J]. Insurance:Mathematics and Economics,2000,26(2-3):133-149.
[27]Wu, X., Yuen, K.C. A discrete-time risk model with interaction between classes of busi-ness[J]. Insurance:Mathematics and Economics,2005,33,117-133.
[28]Ambagaspitiya, R.S. On the distribution of two classes of correlated aggregate claims[J]. Insurance:Mathematics and Economics,1999,24:301-308.
[29]Grandell, J. Aspects of risk theory[M]. New York:Springer-Verlag,1991.
[30]Wang, R.M., Xu, L., Yao, D.J. Ruin problem with stochastic premium stochastic return on investments[J]. Front. Math. China,2007,2(3):467-490.
[31]Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt and Jozef Teugels. Stochastic processes for insurance and finance[M]. New York,1999.
[2]Cramer, H. On the mathematical theory of risk[M]. Stoockholm:Skandia Jubilee Volume, 1930.
[3]Cramer, H. Collective risk theory[M]. Stockholm:Skandia Jubilee Volume,1985.
[4]Gerber, H.U.著.成世学,严颖译.数学风险论导引[M].北京：世界图书出版公司,1997.
[5]成世学.破产论研究综述[J].数学进展,2002,31(5)：403-422.
[6]王广华,吕玉华.保费收入随机化的风险模型[J].经济数学,2006,23(3)：221-228.
[7]Chan, W.-S., Yang, H., Zhang, L. Some results on ruin probabilities in a two-dimensional risk model[J]. Insurance:Mathematics and Economics,2003,32:345-358.
[8]Dang, L., Zhu, N., Zhang, H. Survival probability for a two-dimensional risk model[J]. Insurance:Mathematics and Economics,2009,44(3):491-496.
[9]Li, J., Liu, Z., Tang, Q. On the ruin probabilities of a bidimensional perturbed risk model[J] Insurance:Mathematics and Economics,2007,41:185-195.
[10]Yuen, K.C., Guo, J.Y., Wu, X.Y. On the first time of ruin in the bivariate compound Poisson model[J] Insurance:Mathematics and Economics,2006,38(2):298-308.
[11]王泉,张奕.考虑常利率的二维离散风险模型的破产概率[J].浙江大学学报(理学版),2008,35(5)：501-506.
[12]候丽娟.一类二维风险模型的破产概率[J].泰山学院报,2008,30(6)：48-52.
[13]马学思,刘次华.双Poisson二维风险模型的破产概率[J].统计与决策,2006,(16)：24-25.
[14]李祥发,宋会杰,郭鹏江.扰动的二维风险模型的破产概率[J].西北大学学报(自然科学版),2008,38(6)：885-888.
[15]刘立国,王炳昌,龙国军.重尾索赔下二维风险模型有限时间的破产概率[J].数学理论与应用,2007,27(2)：67-70.
[16]马学思,刘次华.二维相关风险模型的破产概率[J].统计与决策,2009,(2)：151-152.
[17]张旭.具有两类相关索赔的二元风险模型的若干问题[J].重庆工学院报(自然科学版),2008,22(3)：148-152.
[18]Fima C Klebaner. Introduction to stochastic calculus with applications[M]. London:Im-perial College Press,1998.
[19]Nelsen, R.B. An introduction to copulas[M]. Second ed., In:Springer-Series in Statistics, Springer-Verlag, New York,2006.
[20]Kotz, S., Balakrishnan, N., Johnson, N.L. Continuous multivariate distribution, Vol.1. Models and applications[M]. Second ed., Wiley-Interscience, New York,2000.
[21]Cossette, H., Marceau, E., Marri, F. On the compound Poisson risk model with dependence. based on a generalized Farlie-Gumbel-Morgenstern copula[J]. Insurace:Mathematics and Economics,2008,43:444-455.
[22]Drouet mari, D., Kotz, S. Correlation and dependence[M]. Imperial College Press,2001.
[23]Gerber, H.U., Shiu, E.S.W. On the time value of ruin[J]. North American Acturial Journal, 1998,30:48-78.
[24]Willmot, G.E., Lin. Lundberg approximations for compound distributions with insurance application[M]. Springer-Verlag, New York,2001.
[25]Yuen, K.C., Guo, J.Y., Wu, X.Y. On a correlated aggregate claims model with Poisson and Erlang risk process[J]. Insurance:Mathematics and Economics,2002,31:205-214.
[26]Cossette, H., Marceau, E. The discrete-time risk model with correlate classes of business[J]. Insurance:Mathematics and Economics,2000,26(2-3):133-149.
[27]Wu, X., Yuen, K.C. A discrete-time risk model with interaction between classes of busi-ness[J]. Insurance:Mathematics and Economics,2005,33,117-133.
[28]Ambagaspitiya, R.S. On the distribution of two classes of correlated aggregate claims[J]. Insurance:Mathematics and Economics,1999,24:301-308.
[29]Grandell, J. Aspects of risk theory[M]. New York:Springer-Verlag,1991.
[30]Wang, R.M., Xu, L., Yao, D.J. Ruin problem with stochastic premium stochastic return on investments[J]. Front. Math. China,2007,2(3):467-490.
[31]Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt and Jozef Teugels. Stochastic processes for insurance and finance[M]. New York,1999.