基于比例再保险和线性分红策略下风险模型的分析
|
摘要
经典风险模型以及各类推广的风险模型,一般都是以破产概率的一些变动性特征作为理论依据,但是研究发现绝对破产概率才是保险公司更好的预防和控制破产的手段.此外,保险风险模型中的分红策略作为当前研究的热点之一,也得到了人们越来越多的关注.基于再保策略、分红策略、期望折现罚金函数及绝对破产概率等风险理论所得到的研究成果,本文主要从以下三个方面对破产问题进行研究.
首先,介绍风险理论的概念、发展历程、目前主要的发展方向及研究方法,并且利用期望折现罚金函数的定义和性质对基本风险模型的相关性质做了较为全面深入的总结.
其次,作为本文核心之一,在经典风险模型的基础上,引入比例再保险,建立了带干扰的比例再保险风险模型.以更新定理为工具,结合经典风险理论,对该风险模型所涉及的盈余过程的统计特性作了简单分析和推导,给出索赔额服从指数分布时的破产概率和调节系数的显式表达式,求解破产概率的方法与传统方法不同,本文通过生存概率求得了相应风险模型的破产概率.之后,对所得结果进行数值举例,以最大调节系数和最小破产概率作为最优准则,得到了最优再保险策略,而且发现这两类最优准则是等价的.
最后,作为本文的另一核心,通过考虑贷款和投资,建立了具有线性分红策略且带干扰的绝对风险模型.该模型以一个期望折现罚金函数为基础,得到了更具有实际意义的绝对破产概率.此模型的盈余过程是一马氏过程,通过利用其马氏性和全概率公式,给出了关于Gerber-Shiu函数的积分-微分方程,然后结合期望折现罚金函数在特定条件下的具体意义,得到了索赔额服从指数分布时的绝对破产概率和绝对破产时间的Laplace变换.本文较为完整地解决了具有分红策略的绝对破产问题.
For the classical risk model and various generalized risk model, changeablecharacteristics of the ruin probability are the theoretical basis for the insurancecompany, but we find absolute ruin probability is of great significance in reality. Inaddition, the dividends strategies in insurance risk model, as one of the hot spots inthe current research, has also got more and more attention. Based on the research ofthe reinsurance strategy, dividends strategy, expected discounted penalty functionand absolute ruin probability, the structure of this essay is arranged as follows.
Firstly, we introduce the concept, development course, the main developmentdirection and research methods of risk theory at present, and summarize the relevantproperties of the basic risk model thorough the definition and properties of theexpected discounted penalty function in this paper.
Secondly, as one core of this thesis, based on the classical risk model, we take inthe proportion reinsurance then establish the Proportional reinsurance risk modelwith interference. By combining the update theorem with the classical risk the-ory, the statistical properties involved the surplus process are simply analyzed andderived. The specific expression of the ruin probability and the adjustment coe?-cient are given when the amount of claims obey the Exponential distribution.Themethod of Solving the ruin probability is di?erent from the traditional method, andwe obtain the ruin probability corresponding risk model through survival probabilityin this article. After that, some numerical examples are presented to show that theoptimal reinsurance strategies to maximize the adjustment coe?cient and minimizesthe ruin probability, while we find these two criteria are equivalent.
Finally, as another core of this paper, by considering loans and investment, weestablish the absolute risk model with the linear dividends strategy and interference.The model process is a Markov process, using the Markov property and the totalprobability formula, we give a discount of the integral - di?erential equation onthe Gerber-Shiu function; Then according to specific meanings of the Gerber-Shiufunction under certain conditions, The absolute ruin probability and the Laplacetransform of the absolute ruin time are given when the amount of claims obey theExponential distribution.
首先,介绍风险理论的概念、发展历程、目前主要的发展方向及研究方法,并且利用期望折现罚金函数的定义和性质对基本风险模型的相关性质做了较为全面深入的总结.
其次,作为本文核心之一,在经典风险模型的基础上,引入比例再保险,建立了带干扰的比例再保险风险模型.以更新定理为工具,结合经典风险理论,对该风险模型所涉及的盈余过程的统计特性作了简单分析和推导,给出索赔额服从指数分布时的破产概率和调节系数的显式表达式,求解破产概率的方法与传统方法不同,本文通过生存概率求得了相应风险模型的破产概率.之后,对所得结果进行数值举例,以最大调节系数和最小破产概率作为最优准则,得到了最优再保险策略,而且发现这两类最优准则是等价的.
最后,作为本文的另一核心,通过考虑贷款和投资,建立了具有线性分红策略且带干扰的绝对风险模型.该模型以一个期望折现罚金函数为基础,得到了更具有实际意义的绝对破产概率.此模型的盈余过程是一马氏过程,通过利用其马氏性和全概率公式,给出了关于Gerber-Shiu函数的积分-微分方程,然后结合期望折现罚金函数在特定条件下的具体意义,得到了索赔额服从指数分布时的绝对破产概率和绝对破产时间的Laplace变换.本文较为完整地解决了具有分红策略的绝对破产问题.
For the classical risk model and various generalized risk model, changeablecharacteristics of the ruin probability are the theoretical basis for the insurancecompany, but we find absolute ruin probability is of great significance in reality. Inaddition, the dividends strategies in insurance risk model, as one of the hot spots inthe current research, has also got more and more attention. Based on the research ofthe reinsurance strategy, dividends strategy, expected discounted penalty functionand absolute ruin probability, the structure of this essay is arranged as follows.
Firstly, we introduce the concept, development course, the main developmentdirection and research methods of risk theory at present, and summarize the relevantproperties of the basic risk model thorough the definition and properties of theexpected discounted penalty function in this paper.
Secondly, as one core of this thesis, based on the classical risk model, we take inthe proportion reinsurance then establish the Proportional reinsurance risk modelwith interference. By combining the update theorem with the classical risk the-ory, the statistical properties involved the surplus process are simply analyzed andderived. The specific expression of the ruin probability and the adjustment coe?-cient are given when the amount of claims obey the Exponential distribution.Themethod of Solving the ruin probability is di?erent from the traditional method, andwe obtain the ruin probability corresponding risk model through survival probabilityin this article. After that, some numerical examples are presented to show that theoptimal reinsurance strategies to maximize the adjustment coe?cient and minimizesthe ruin probability, while we find these two criteria are equivalent.
Finally, as another core of this paper, by considering loans and investment, weestablish the absolute risk model with the linear dividends strategy and interference.The model process is a Markov process, using the Markov property and the totalprobability formula, we give a discount of the integral - di?erential equation onthe Gerber-Shiu function; Then according to specific meanings of the Gerber-Shiufunction under certain conditions, The absolute ruin probability and the Laplacetransform of the absolute ruin time are given when the amount of claims obey theExponential distribution.
引文
[1] Lundberg F I. I.Approximerad framstallning av scannolikhets function.II, Atersfor-sakring av kollektiveriske. Uppsala:Almqvist&Wiksell, 1903
[2] Cramer H. On the mathematical theory of risk. Stockholm:Scandia Jubilee Volume,1930
[3]龚日朝,扬向群.复合二项风险模型下的破产概率.吉首大学学报, 2000, 21(4):41-43
[4] Jan G. Aspects of risk theory. Spring-Verlag New York, 1991
[5]王绍锋.关于Erlang(n)风险过程的破产概率.贵州师范大学学报, 2006, 24(2): 74-77
[6]罗建华,宋熠.一个风险模型的生存概率.中南林业科技大学学报, 2009, 29(3): 138-141
[7] Dufresne F, Gerber H. Risk theory for the compound Poisson process that is perturedby di?usion. Insurance: Mathematics and Economics, 1991, (10), 51-59
[8]江涛.具有扩散项的停止—损失再保险问题的破产概率.中国管理科学, 2006, 14(6):11-15
[9]孙树旺,江涛.具有扩散项的带免赔额的Erlang(2)问题的破产概率.中国管理科学, 2007, 15(5): 36-41
[10] Dickson D C M, Waters H R. Reinsurance and ruin. Insurance: Mathematics andEconomics, 1996, 19: 61-80
[11] Dickson D C M, Waters H R. Relative reinsurance retention levels. Insurance: Math-ematics and Economics, 1997, 7: 207-227
[12] Hojgaard B, Taksar M. Optimal proportional reinsurance policies for di?usion models.Scand Actuarial Journal, 1998, 7: 166-180
[13] Schmidli H. Optimal proportional reinsurance policies in a dynamic setting. ScandActuarial Journal, 2001, 15: 55-68
[14] Schmidli H. On minimizing the ruin probality by investment and reinsurance. Annalsof Applied Probability, 2002, 2(3): 890-907
[15] Hipp C, Plum M. Optimal investment for insurers. Insurance: Mathematics and Eco-nomics, 2000, 27: 215-228
[16] Dickson D C M, Reis A D. The e?ect of interest on negative surplus. Insurance:Mathematics and Economics, 1997, 21: l-16
[17] Dassios A, Embrechts E. Martingales and insurance risk. Communications in stat-ics―stochastic models, 1989, 5: 118-127
[18] David L. Constant dividend barrier in a risk model with inter claim―dependent claimsizes. Insurance: Mathmatics and Economics, 2008, 42: 31-38
[19] Yuan H L, Hu Y J. Yuan H L, Hu Y J. Absolute ruin in the compound Poisson riskmodel with constant dividend barrier. Stachastics and Probability Letters, 2008, 78:2086-2094
[20] Lin X S, Pavlova K P. Lin X S, Pavlova K P. The compound Poisson risk model witha theshold dividend strategy. Insurance: Mathematics and Economics, 2006, 38: 57-80
[21]徐沈新,方大凡.破产理论研究的历史与现状.吉首大学学报, 2008, 29: 29-33
[22]宗昭君,胡蜂,元春梅.具有线性红利界限的破产理论.工程数学学报, 2006, 23(2):319-323
[23]成世学,严颖.数学风险论导引.北京:世界图书出版公司, 1996, 10-55
[24] Gerber H U, Shiu E S W. On the time value of ruin. North American ActuarialJournal, 1998, 2: 48-78
[25] Gerber H U, Shiu E S W. The joint distribution of the time of ruin, the surplus im-mediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics,1997, 21: 129-137
[26] Cai J, Dickson D C M. On the expected discounted penalty functions at ruin of asurplus process with interest. Insurance: Mathematics and Economics, 2002, 30: 389-404
[27] Yao D G, Wang R G, Xu L. On the expected discounted penalty functions associatedwith the time of ruin for a risk model with random income. Chinese Journal of AppliedProbabilities and Statistics, 2008, 6: 319-326
[28] Wu R, Li W. The probability of ruin in a kind of Cox risk model with variablepremium rate. Scand Actuarial Journal, 2004, 2: 121-132
[29]张燕,田铮,刘向增.具有二阶保费率的Erlang(2)风险过程.工程数学学报, 2009,26(3): 443-450
[30]周明,张春生.古典风险模型下的绝对破产.应用数学学报, 2005, 28: 695-703
[31] Willmot G E, Dickson D C M. The Gerber-Shiu discounted penalty function in thesationary renewal risk model. Insurance: Mathematics and Economics, 2003, 32: 403-411
[32] Gerber H U, Yang H L. Absolute ruin probabilities in jump di?usion risk model withinvestment. Insurance: Mathematics and Economics, 2007, 11(3): 159-169
[33] Zhan X B, Wang L Z, Zhang C S. The Gerber-shiu discounted penalty function ofthe classical absolute ruin model with investment and loan. Journal of Tianjin NormalUniversity, 2009, 29(1): 16-19
[34] Yuen K C, Wang G J, Li W K. The Gerber-shiu expected discounted penalty functionfor risk process with interest and a constant dividend barrier. Insurance: Mathematicsand Economics, 2007, 40, 104-112
[35]李文婷.阈红利策略下复合Poisson风险模型的绝对破产: [兰州大学硕士学位论文].兰州:兰州大学应用数学, 2010, 5-22
[36] Cai J. On the time value of absolute with debit interest. Advances in Applied Prob-ability, 2007, 39: 343-359
[37] Lin X S, Willmot G E, Drekic S. The classical risk model with a constant dividend bar-rier: analysis of the Gerber-Shiu discounted penalty function. Insurance:Mathematicsand Economics, 2003, 33: 551-566
[38] Bao Z H, Ye Z X. The Gerber―Shiu discounted penalty function in the delayedrenewal risk process with random income. Applied Mathematics and Computation,2007, 84: 857-863
[39]黎锁平,刘坤会.随机控制的方程与证券投资模型.北方交通大学学报, 2003,27(6): 63-67
[40] Yang H, Zhang L. On the distribution of surplus immediately after ruin under interestforce.Insurance: Mathematics and Economics, 2001, 29: 247-255
[41] Bremaud P. A martingale approach to point processes. Ph..D.Dissertation, The Uni-versity of Califormia, Berkeley, 1972, 2: l-45
[42] Bremaud P. The martingale theory of point processes over the real half line. Lect.Notes in Economics and Mathematical systems, 1974, 107: 519-542
[43] Bremaud P. Point Processes and Queues:Martingale Dynamics. Rlin: Springer-Verlag, 1981, 22-36
[44]黎锁平,刘琪.投资和干扰具有随机保费的离散风险模型.高校应用数学学报,2009, 24(1): 9-14
[45]裴萌.按比例分红策略下带有常利率的复合Poisson-Geometric风险模型――Gerber-Shiu折现罚金函数: [南开大学硕士学位论文].天津:南开大学概率论与数理统计, 2009, 14-31
[46]苏桂春.几类阈红利边界策略下Gerber-Shiu罚金折现函数研究: [兰州大学硕士学位论文].兰州:兰州大学数学与应用数学, 2010, 5-22
[2] Cramer H. On the mathematical theory of risk. Stockholm:Scandia Jubilee Volume,1930
[3]龚日朝,扬向群.复合二项风险模型下的破产概率.吉首大学学报, 2000, 21(4):41-43
[4] Jan G. Aspects of risk theory. Spring-Verlag New York, 1991
[5]王绍锋.关于Erlang(n)风险过程的破产概率.贵州师范大学学报, 2006, 24(2): 74-77
[6]罗建华,宋熠.一个风险模型的生存概率.中南林业科技大学学报, 2009, 29(3): 138-141
[7] Dufresne F, Gerber H. Risk theory for the compound Poisson process that is perturedby di?usion. Insurance: Mathematics and Economics, 1991, (10), 51-59
[8]江涛.具有扩散项的停止—损失再保险问题的破产概率.中国管理科学, 2006, 14(6):11-15
[9]孙树旺,江涛.具有扩散项的带免赔额的Erlang(2)问题的破产概率.中国管理科学, 2007, 15(5): 36-41
[10] Dickson D C M, Waters H R. Reinsurance and ruin. Insurance: Mathematics andEconomics, 1996, 19: 61-80
[11] Dickson D C M, Waters H R. Relative reinsurance retention levels. Insurance: Math-ematics and Economics, 1997, 7: 207-227
[12] Hojgaard B, Taksar M. Optimal proportional reinsurance policies for di?usion models.Scand Actuarial Journal, 1998, 7: 166-180
[13] Schmidli H. Optimal proportional reinsurance policies in a dynamic setting. ScandActuarial Journal, 2001, 15: 55-68
[14] Schmidli H. On minimizing the ruin probality by investment and reinsurance. Annalsof Applied Probability, 2002, 2(3): 890-907
[15] Hipp C, Plum M. Optimal investment for insurers. Insurance: Mathematics and Eco-nomics, 2000, 27: 215-228
[16] Dickson D C M, Reis A D. The e?ect of interest on negative surplus. Insurance:Mathematics and Economics, 1997, 21: l-16
[17] Dassios A, Embrechts E. Martingales and insurance risk. Communications in stat-ics―stochastic models, 1989, 5: 118-127
[18] David L. Constant dividend barrier in a risk model with inter claim―dependent claimsizes. Insurance: Mathmatics and Economics, 2008, 42: 31-38
[19] Yuan H L, Hu Y J. Yuan H L, Hu Y J. Absolute ruin in the compound Poisson riskmodel with constant dividend barrier. Stachastics and Probability Letters, 2008, 78:2086-2094
[20] Lin X S, Pavlova K P. Lin X S, Pavlova K P. The compound Poisson risk model witha theshold dividend strategy. Insurance: Mathematics and Economics, 2006, 38: 57-80
[21]徐沈新,方大凡.破产理论研究的历史与现状.吉首大学学报, 2008, 29: 29-33
[22]宗昭君,胡蜂,元春梅.具有线性红利界限的破产理论.工程数学学报, 2006, 23(2):319-323
[23]成世学,严颖.数学风险论导引.北京:世界图书出版公司, 1996, 10-55
[24] Gerber H U, Shiu E S W. On the time value of ruin. North American ActuarialJournal, 1998, 2: 48-78
[25] Gerber H U, Shiu E S W. The joint distribution of the time of ruin, the surplus im-mediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics,1997, 21: 129-137
[26] Cai J, Dickson D C M. On the expected discounted penalty functions at ruin of asurplus process with interest. Insurance: Mathematics and Economics, 2002, 30: 389-404
[27] Yao D G, Wang R G, Xu L. On the expected discounted penalty functions associatedwith the time of ruin for a risk model with random income. Chinese Journal of AppliedProbabilities and Statistics, 2008, 6: 319-326
[28] Wu R, Li W. The probability of ruin in a kind of Cox risk model with variablepremium rate. Scand Actuarial Journal, 2004, 2: 121-132
[29]张燕,田铮,刘向增.具有二阶保费率的Erlang(2)风险过程.工程数学学报, 2009,26(3): 443-450
[30]周明,张春生.古典风险模型下的绝对破产.应用数学学报, 2005, 28: 695-703
[31] Willmot G E, Dickson D C M. The Gerber-Shiu discounted penalty function in thesationary renewal risk model. Insurance: Mathematics and Economics, 2003, 32: 403-411
[32] Gerber H U, Yang H L. Absolute ruin probabilities in jump di?usion risk model withinvestment. Insurance: Mathematics and Economics, 2007, 11(3): 159-169
[33] Zhan X B, Wang L Z, Zhang C S. The Gerber-shiu discounted penalty function ofthe classical absolute ruin model with investment and loan. Journal of Tianjin NormalUniversity, 2009, 29(1): 16-19
[34] Yuen K C, Wang G J, Li W K. The Gerber-shiu expected discounted penalty functionfor risk process with interest and a constant dividend barrier. Insurance: Mathematicsand Economics, 2007, 40, 104-112
[35]李文婷.阈红利策略下复合Poisson风险模型的绝对破产: [兰州大学硕士学位论文].兰州:兰州大学应用数学, 2010, 5-22
[36] Cai J. On the time value of absolute with debit interest. Advances in Applied Prob-ability, 2007, 39: 343-359
[37] Lin X S, Willmot G E, Drekic S. The classical risk model with a constant dividend bar-rier: analysis of the Gerber-Shiu discounted penalty function. Insurance:Mathematicsand Economics, 2003, 33: 551-566
[38] Bao Z H, Ye Z X. The Gerber―Shiu discounted penalty function in the delayedrenewal risk process with random income. Applied Mathematics and Computation,2007, 84: 857-863
[39]黎锁平,刘坤会.随机控制的方程与证券投资模型.北方交通大学学报, 2003,27(6): 63-67
[40] Yang H, Zhang L. On the distribution of surplus immediately after ruin under interestforce.Insurance: Mathematics and Economics, 2001, 29: 247-255
[41] Bremaud P. A martingale approach to point processes. Ph..D.Dissertation, The Uni-versity of Califormia, Berkeley, 1972, 2: l-45
[42] Bremaud P. The martingale theory of point processes over the real half line. Lect.Notes in Economics and Mathematical systems, 1974, 107: 519-542
[43] Bremaud P. Point Processes and Queues:Martingale Dynamics. Rlin: Springer-Verlag, 1981, 22-36
[44]黎锁平,刘琪.投资和干扰具有随机保费的离散风险模型.高校应用数学学报,2009, 24(1): 9-14
[45]裴萌.按比例分红策略下带有常利率的复合Poisson-Geometric风险模型――Gerber-Shiu折现罚金函数: [南开大学硕士学位论文].天津:南开大学概率论与数理统计, 2009, 14-31
[46]苏桂春.几类阈红利边界策略下Gerber-Shiu罚金折现函数研究: [兰州大学硕士学位论文].兰州:兰州大学数学与应用数学, 2010, 5-22