EUT与最优再保险：基于整体市场的研究
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摘要
期望效用理论(EUT)自从得到NM公理化体系方法之后,其在经济理论中的应用研究得到极大的关注和深入,在金融保险领域的应用研究也是莫能例外.上个世纪五十年代以来,EU理论得到广泛深入的发展,有一系列的概念体系和成熟的理论框架,有Arrow-Praat的风险厌恶系数和最大期望效用模型(MMEU)以及Yaari的广义期望效用理论等.MMEU模型主要用于分析研究不确定性风险环境下风险决策者行为如何达到最大化的效用.
本文借助于效用理论工具,基于保险参与者(保险公司和再保险公司双方)都是风险厌恶的,在考虑双方的整体保险市场均衡理论框架下,分析保险和再保险双方的最优保险与再保险策略.
传统上,从保险公司来说,如果不考虑自身的风险,基于保险公司的风险盈余过程,一般的模型是为保险公司面临的索赔总额,p(t)为时刻t之前的保费收入
考虑到传统的保险再保险理论分析框架基本上都是从保险方的角度展开分析的,而且多数是考虑一家保险公司,也没有考虑到再保险方的利益和收益.实际上,保险市场存在多方利益的均衡和风险的分散与共担,因此本文将研究的视角和重点转到再保险方,同时考虑到保险公司和再保险公司都拥有可保的巨额风险.也就是,保险方和再保险方都具有保险和再保险的双重身份,每一家公司都既有分出再保险,也有分入再保险——这是近年来保险业发展的新趋势!——本文借此展开深入的分析研究,将传统的经典风险模型推广到更为一般的切合实际联合共保环境的状态,建立了多家保险公司(都拥有巨额风险)联合共保的系统化保险市场模型.
第一章概要介绍本文的研究思路和主要研究成果以及主要的创新之处.本文基于MMEU和风险过程理论,将保险和在保险置于一个金融保险市场(Verbund)之下加以分析展开,主要得到巨型风险保险和再保险市场的联合分保模型下的破产概率等结果.
第二章介绍有关效用理论的基本概念和理论框架,以及在EUT基础之上保险和再保险的风险收益(保费)性质和常用的保费原则.
第三章介绍了不确定性风险环境下的保险和再保险原理,主要讨论基本的再保险类型和常见的再保险形式,并给出了初步的比较分析.
第四章从Arrow、Borch和Gerber等人的结果出发,对于风险厌恶的参与者组成的保险和再保险市场,基于保险参与者各方的最大效用分析框架,给出了多家保险公司联合体下整个保险市场联合共保的系统模型,并介绍了基于MMEU原理下保险与再保险均衡最优的分析框架.
第五章则从MMEU的角度,针对复合Poisson过程的指数效用函数和指数索赔情形,给出了比例再保险和非比例再保险(XLR和SLR)等各种类别的情形下有关保险方和再保险方的风险溢价的上下界.
第六章侧重研究巨额索赔下的保险与再保险的顺序统计量(X(1),X(2),X(3),…,X(N))的分布和矩计算等问题,着重研究LCR和ECOMOR下
以及的有关结果.
第七章进一步研究基于整体风险保险环境下联合再保险的几个概念及其初步的性质,并就比例分保再保险情形给出了各家保险公司的调节系数和破产概率的推导,同时给出了风险转移系数、风险转移矩阵的数学表示以及分数效应的经济学初步解释.
Since Expected Utility theory (EUT for short.) was axiomatized by Von Neumann and Oskar Morgenstern in 1940's, its applications in economics and practice are given much more concerns and deep investigation. From 1950's on, EUT has been developed extensively and thereof a set of concepts and theoretic analysis frames come down such as Arrow-Pratt's Risk aversion and MMEU, as well as Yaari's generalized expected utility theory in 1980's and so on. MMEU model is mainly used to analyze how decision makers act to maximize the effectiveness of risk and uncertainty in a risky environment.
In the context of this thesis, both insurers and reinsurers are regarded as risk-averse, and they both make up the overall insurance Verbund. Based on this assumption, the optimization of the decisions of both insurers and reinsurers is done by means of EUT.
As known, from the point of view of the insurance company, and regardless of its own insurable risks, its risk surplus process goes as follows: Here S(t)=(?) is the aggregate claims of the insurer, and p(t) is the total premium income until the time t.
From the model above, we can find without much difficulty that traditional consideration on the insurance and reinsurance is expanded only from unilateral, that is, either only from the insurer's view or from the reinsurer's alone. More often than not, almost all discussions take only the insurer into account, and so the insured's are ignored;So is the situation of the insurer and the reinsurer, attention is more paid on one and the other is lost in sight. In fact, the insurance is an economic behavior done by both the insurer and the insured, or by the insurer and the reinsurer. And that is not the all. It is indisputable that the insurance Verbund, as a whole, is a Game-equilibrium or the balance of bargaining by all sides of the market. In the sight of this view, we take both sides of the insurer and the reinsurer into account. And so, whatever the insurer is of outward reinsurance or inward reinsurance, any insurance company ought to share all risks in the insurance Verbund. From this point of view, any insurance company reinsurance outward and inward meanwhile. Any party of the market would cede its risks to others, and retrocede the risks from other companies. As a result of this view, a new model of systemic co-insurance by n insurance companies is developed in Chapter 3 based on classical L-C risk surplus process. At the end of this thesis, the concepts of risk transferring coefficients and risk transferring matrix are put out. Details will be shown below in the corresponding chapters and sections respectively.
In Chapter 2, an available glimpse of the fundamental frame of EUT is introduced to put forward some premium principles as well as their properties.
In Chapter 3, the general theories are brought out about insurance and reinsurance in the risky or uncertainty environment. Basic types, forms and functions of reinsurance are advanced here, especially about quate share reinsurance, no proportional reinsurances such as XLR, SLR, Co-insurance, LCR and ECOMOR, etc.
In Chapter 4, based on the work of Arrow's、Borch's and Gerber's, all risk-averse parts of the insurance Verbund have the same objective of MMEU. A systemic co-insurance model of many insurers and reinsurers is introduced along with the analysis approach out of Borch's insurance market equilibrium theory built on MMEU. In this context, no matter what is the company in the insurance Verbund it is taken as an insurance company and a reinsurance one again.
In Chapter 5, from the perspective of MMEU, we derive the lower and upper bounds of risk premiums of both the insurer and the re-insurer if they are both of exponential utility functions with exponential individual claims in compound Poisson process according to proportional reinsurance and XLR and SLR respectively.
In Chapter 6, we give some preliminaries about large claims reinsurance and ECOMOR.
In the last Chapter, concentration is put on a simplified co-insurance model in which we focus on proportional co-reinsurance with exponential individual claims in compound Poisson process and then the ruin probability of the ith company is obtained without much trouble, and thereby Risk Transferring Coefficients and Risk Transferring Matrix, and Fractional Effect are proposed as very useful cocepts.
本文借助于效用理论工具,基于保险参与者(保险公司和再保险公司双方)都是风险厌恶的,在考虑双方的整体保险市场均衡理论框架下,分析保险和再保险双方的最优保险与再保险策略.
传统上,从保险公司来说,如果不考虑自身的风险,基于保险公司的风险盈余过程,一般的模型是为保险公司面临的索赔总额,p(t)为时刻t之前的保费收入
考虑到传统的保险再保险理论分析框架基本上都是从保险方的角度展开分析的,而且多数是考虑一家保险公司,也没有考虑到再保险方的利益和收益.实际上,保险市场存在多方利益的均衡和风险的分散与共担,因此本文将研究的视角和重点转到再保险方,同时考虑到保险公司和再保险公司都拥有可保的巨额风险.也就是,保险方和再保险方都具有保险和再保险的双重身份,每一家公司都既有分出再保险,也有分入再保险——这是近年来保险业发展的新趋势!——本文借此展开深入的分析研究,将传统的经典风险模型推广到更为一般的切合实际联合共保环境的状态,建立了多家保险公司(都拥有巨额风险)联合共保的系统化保险市场模型.
第一章概要介绍本文的研究思路和主要研究成果以及主要的创新之处.本文基于MMEU和风险过程理论,将保险和在保险置于一个金融保险市场(Verbund)之下加以分析展开,主要得到巨型风险保险和再保险市场的联合分保模型下的破产概率等结果.
第二章介绍有关效用理论的基本概念和理论框架,以及在EUT基础之上保险和再保险的风险收益(保费)性质和常用的保费原则.
第三章介绍了不确定性风险环境下的保险和再保险原理,主要讨论基本的再保险类型和常见的再保险形式,并给出了初步的比较分析.
第四章从Arrow、Borch和Gerber等人的结果出发,对于风险厌恶的参与者组成的保险和再保险市场,基于保险参与者各方的最大效用分析框架,给出了多家保险公司联合体下整个保险市场联合共保的系统模型,并介绍了基于MMEU原理下保险与再保险均衡最优的分析框架.
第五章则从MMEU的角度,针对复合Poisson过程的指数效用函数和指数索赔情形,给出了比例再保险和非比例再保险(XLR和SLR)等各种类别的情形下有关保险方和再保险方的风险溢价的上下界.
第六章侧重研究巨额索赔下的保险与再保险的顺序统计量(X(1),X(2),X(3),…,X(N))的分布和矩计算等问题,着重研究LCR和ECOMOR下
以及的有关结果.
第七章进一步研究基于整体风险保险环境下联合再保险的几个概念及其初步的性质,并就比例分保再保险情形给出了各家保险公司的调节系数和破产概率的推导,同时给出了风险转移系数、风险转移矩阵的数学表示以及分数效应的经济学初步解释.
Since Expected Utility theory (EUT for short.) was axiomatized by Von Neumann and Oskar Morgenstern in 1940's, its applications in economics and practice are given much more concerns and deep investigation. From 1950's on, EUT has been developed extensively and thereof a set of concepts and theoretic analysis frames come down such as Arrow-Pratt's Risk aversion and MMEU, as well as Yaari's generalized expected utility theory in 1980's and so on. MMEU model is mainly used to analyze how decision makers act to maximize the effectiveness of risk and uncertainty in a risky environment.
In the context of this thesis, both insurers and reinsurers are regarded as risk-averse, and they both make up the overall insurance Verbund. Based on this assumption, the optimization of the decisions of both insurers and reinsurers is done by means of EUT.
As known, from the point of view of the insurance company, and regardless of its own insurable risks, its risk surplus process goes as follows: Here S(t)=(?) is the aggregate claims of the insurer, and p(t) is the total premium income until the time t.
From the model above, we can find without much difficulty that traditional consideration on the insurance and reinsurance is expanded only from unilateral, that is, either only from the insurer's view or from the reinsurer's alone. More often than not, almost all discussions take only the insurer into account, and so the insured's are ignored;So is the situation of the insurer and the reinsurer, attention is more paid on one and the other is lost in sight. In fact, the insurance is an economic behavior done by both the insurer and the insured, or by the insurer and the reinsurer. And that is not the all. It is indisputable that the insurance Verbund, as a whole, is a Game-equilibrium or the balance of bargaining by all sides of the market. In the sight of this view, we take both sides of the insurer and the reinsurer into account. And so, whatever the insurer is of outward reinsurance or inward reinsurance, any insurance company ought to share all risks in the insurance Verbund. From this point of view, any insurance company reinsurance outward and inward meanwhile. Any party of the market would cede its risks to others, and retrocede the risks from other companies. As a result of this view, a new model of systemic co-insurance by n insurance companies is developed in Chapter 3 based on classical L-C risk surplus process. At the end of this thesis, the concepts of risk transferring coefficients and risk transferring matrix are put out. Details will be shown below in the corresponding chapters and sections respectively.
In Chapter 2, an available glimpse of the fundamental frame of EUT is introduced to put forward some premium principles as well as their properties.
In Chapter 3, the general theories are brought out about insurance and reinsurance in the risky or uncertainty environment. Basic types, forms and functions of reinsurance are advanced here, especially about quate share reinsurance, no proportional reinsurances such as XLR, SLR, Co-insurance, LCR and ECOMOR, etc.
In Chapter 4, based on the work of Arrow's、Borch's and Gerber's, all risk-averse parts of the insurance Verbund have the same objective of MMEU. A systemic co-insurance model of many insurers and reinsurers is introduced along with the analysis approach out of Borch's insurance market equilibrium theory built on MMEU. In this context, no matter what is the company in the insurance Verbund it is taken as an insurance company and a reinsurance one again.
In Chapter 5, from the perspective of MMEU, we derive the lower and upper bounds of risk premiums of both the insurer and the re-insurer if they are both of exponential utility functions with exponential individual claims in compound Poisson process according to proportional reinsurance and XLR and SLR respectively.
In Chapter 6, we give some preliminaries about large claims reinsurance and ECOMOR.
In the last Chapter, concentration is put on a simplified co-insurance model in which we focus on proportional co-reinsurance with exponential individual claims in compound Poisson process and then the ruin probability of the ith company is obtained without much trouble, and thereby Risk Transferring Coefficients and Risk Transferring Matrix, and Fractional Effect are proposed as very useful cocepts.
引文
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77. Maria de Lourdes Centeno (2002). Measuring the effects of reinsurance by the adjustment coefficient in the Sparre Anderson model Insurance Math. Econom 30. pp37-49.
78. Vernic, R. (1997) On the bivariate generalized Poisson distribution. ASTIN Bulletbl 27, pp23-31.
79. CAI, J. TAN, K. S.(2007). Optimal retention fora stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bull.,37(1), pp93-112.
80. CAI, J.Tan, K. S.Weng, C.& Zhang, Y.(2008). Optimal reinsurance under VaR and CTE risk measures. Insurance Math. Econom.43, pp185-196.
81. Centeno, M. L.(2004). Retention and Reinsurance Programmes, in Encyclopedia of Actuarial Science,3,pp1443-1452.
82. Centeno, M. L. Guerra, M. (2008). The optimal reinsurance strategy-the individual claim case,http://cemapre.iseg.utl.pt/archive/preprints/ORS.pdf
83. Michel Denuit, Catherine Vermandele(1998).Optimal reinsurance and stop-loss order. Insurance Math. Econom 22.pp 229-233.
84. Ladoucette, S.A. Teugels, J.L.(2006). Reinsurance of large claims. Journal of Computational and Applied Mathematics 186 (1), pp163-190.
85. Stigler, G. (1950). The Development of Utility Theory:Ⅰ;Ⅱ. Journal of Political Economy,58,pp307-327; pp373-396.
86. Zaks, Y. Frostig, E. Levikson B. (2006) Optimal pricing of a hetero-geneous portfolio for a given risk level. ASTIN Bulletin,36(1), pp161-185.
87. G.C. Taylor(1985). A heuristic review of some ruin theory results. ASTIN Bulletin 15, pp73-88.
88. Centeno, M.L. (1997), Excess of Loss Reinsurance and the Probability of Ruin in Finite Horizon. ASTIN Bulletin, Vol.27, No1.pp59-70.
89. D.Samson,H.Thomas.(1983):Reinsurance Decision Making and Expected Utility. The Journal of Risk and Insurance, Vol.50, No.2, pp 249-264.
90. Gajek, L. Zagrodny, D.(2004). Optimal reinsurance under general risk measures, Insurance Math.Econom.34.pp105-112.
91. Benktander, G.(1975):A Note on Optimal Reinsurance. ASTIN Bulletin, VIII, pp154-164.
92. Danny Samson A. Wirth (1990). Expected utility strategic decision models for general insurers. Omega, Vo.18, (3), pp259-267.
93. Chris Starmer(2000). Developments in Non-Expected Utility Theory. Journal of Economic Literature. Vol.ⅩⅩⅩⅧ (6). pp332-382.
94. Guerra, M. Centeno, M. L.(2008). Optimal reinsurance policy:The adjustment coefficient and the expected utility criteria, Insurance Math. Econom.,42, pp529-539.
95. Jun Jiang, Qihe Tang. (2008). Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims. Insurance Math. Econom.Vol 43.pp431-436.
96. Lampaert, I. Walhin, J.F. (2005). On the Optimality of Proportional Reinsurance. Scandinavian Actuarial Journal(3),pp 225-239.
97. E.Kremer. (1982) Rating of largest claims and ECOMOR reinsurance treaties for large portfolios, ASTIN Bull.13 (1).pp47-56.
98. E.Kremer,(1990).The asymptotic efficiency of largest claims reinsurance treaties, ASTIN Bull.20 (1).pp11-22.
99. Glineur, F. J.F. Walhin (2006), de Finetti's retention problem for proportional reinsurance revisited, Unpublished manuscript.
100. Hansjorg Albrecher. Jozef L. Teugels.(2008) On excess-of-loss reinsurance. Teor. Tmovir. ta Matem. Statyst.No79, pp5-19.
101. Pesonen, M.I. (1984). Optimal reinsurances. Scandinavian Actuarial Journal 84, pp65-90.
102. Tversky, A.& Kahneman, D. (1991). Loss Aversion in Riskless Choice: A Reference Dependent Model. Quarterly Journal of Economics 106, pp1039-1061.
103. Matthew Rabin R. H. Thaler. (2000). Risk Aversion. Journal of Economic Perspectives. Vol15. pp219-232.
104. Amos Tversky, Daniel Kahneman.(1991)Loss Aversion in Riskless Choice A Reference-Dependent Model. The Quarterly Journal of Economics, Vol.106, No.4. pp 1039-1061.
105. E. Jouini, W. Schachermayer N. Touzi (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance 18. pp269-292.
106. Wang, S. S. V. R. Young (1998), Ordering risks:Expected utility theory versus Yaari's dual theory of risk, Insurance Math. Econom,22 (2): pp145-161.
107. 王梓坤.(1965)随机过程论.科学出版社.
108. 邓永录,梁之舜.(1998)随机点过程及其应用.北京：科学出版社.
109. Gerber.H. U.(1979),成世学,严颖译.数学风险论导引.世界图书出版社.
110. 劳斯.何声武,谢盛荣 等译.(1997)随机过程.北京：中国统计出版社.
111. [荷]R.卡尔斯,M.胡法兹,J.达纳,M.狄尼特著,唐启鹤,胡太忠,成世学译(2005).现代精算风险理论.(北京)科学出版社.
112. 赵晓芹,刘再明.(2008).广义二元复合Poisson风险模型破产概率的估计.统计研究,Vol.25(5),pp94-97.
113. 王晶刚,刘再明,周永卫(2005).保险系统中一类双险种风险模型的破产概率.数学理论与应用, Vol.25(1),pp39-42.
114. 江五元,刘再明.(2009)带阈限分红策略的一类更新风险模型.应用数学学报.Vol.32(3).pp454-466.
115. 赵晓芹,王国宝,刘再明.(2004)广义二元复合非齐次Poisson风险模型的破产概率.长沙理工大学学报(自然科学版)Vol.1.(3-4),pp74-77.
116. 赵晓芹,刘再明.(2004)一类双险种风险过程的破产概率的估计.数学理论与应用,Vol.24(1).pp97-100.
117. 刘东海,彭丹,刘再明.(2009)相依索赔的二项风险模型的破产问题.高校应用数学学报.Vol24(3).pp259-265.
118. 李俊海,刘再明.(2006)双更新风险模型.应用数学学报.Vol.29(3).pp344-351.
119. 刘汉进(2008).具有专属保险特性的国内保险公司发展及其影响.金融与经济,第2期,pp65-67.
120.龚日朝,李凤军(2001).双Poisson风险模型下的破产概率.湘潭师范学院学报(自然科学版)Vol.23.pp55-57.
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11. Borch, K.(1960).The Safety Loading of Reinsurance Premiums. Skandi-navisk Aktuarietidshrift, pp163-184.
12. Borch, K. (1974) The Mathemancal Theory of Insurance. Lexington Books Lexington, Mass.pp25-100.
13. Wyler, E. (1990) Pareto Optimal Risk Exchanges and a system of differential equations:a duality theorem. ASTIN Bulletin,20:23-32.
14. Gerber, H.U. & Pafumi, G. (1998). Utility functions:from risk theory to finance, including discussion, North American Actuarial Journal 2(3), pp74-100..
15. Gerber H. U. (1976). On additive premium calculation principles, ASTIN Bulletin 7(3), pp215-222.
16. Gerber H. U. (1982):A Remark on the Principle of Zero Utility, ASTIN Bulletin,.13(2), pp133-134.
17. Reich, A. (1985). Homogeneous premium calculation principles. ASTIN Bulletin 14, pp123-133.
18. Young, V.R.(1999).Optimal insurance under Wang's premium principle. Insurance Math. Economics.25, pp109-122.
19. Wang S.Young, V.R.Panjer H.H.(1997) Axiomatic characterization of insurance prices. Insurance Math. Economics.21 (2),pp173-183.
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21. Arthur Charpentier.(2008).Optimal reinsurance with ruin probability target. http://resim.irisa.fr/Slides/Friday/slides-irisa.pdf,7th International Work-shop on Rare Event Simulation, Sept.
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23. Borch.H, K. (1962). Equilibrium in a reinsurance market, Econometrica 30, pp424-444.
24. Borch, K. (1962) Reformulation of Some Problems in the Theory of Risk, Pro- ceedings of the Casualty Actuarial. Society,46, pp109-118.
25. Anja De Waegenaere.(1994) Equilibria in a mixed financial-reinsurance market. Insurance:Mathematics and Ecomonics,14. pp205-218.
26. Kenneth J. Arrow; Gerard Debreu.(1954) Existence of an Equilibrium for a Competitive Economy, Econometrica. pp265-290.
27. Borch, K(1986). Risk theory and serendipity. Insurance:Mathematics and Econotntcs 5, pp103-112.
28. Buhlmann, H. (1980). An Economic Premium Principle. ASTIN Bulletin, 11,pp52-60.
29. Buhlmann, H. (1984). The general economic premium principle, ASTIN Bulletin 14(1), pp13-22.
30. Muller, H.H. (1987). Economic premium principles in insurance and the capital asset pricing model, ASTIN Bulletin 17(2), pp141-150.
31. Dickson, D.C.M. Waters, H.R. (1996) Reinsurance and ruin. Insurance Math. Econom. Vol.19, pp61-80.
32. Virginnia R. Young (2002):Premium Principle, Modern Actuarial Risk Theory, pp111-126.
33. Gerber, H. U.(1975). A Geometric Proof of Borch's Theorem, Bulletin of the Swiss Association of Actuaries, pp183-187.
34. Aase, K.(1993). Equilibrium in a reinsurance syndicate; Existence, uniqueness and characterization. ASTIN Bulletin,23, No 2, pp185-211.
35. Pierre-Francois Koehl and Jean-Charles Rochet:Equilibrium in a Reinsurance Market:Introducing Taxes. Geneva Risk Ins Rev.Vol.19: pp101-117.
36. Pesonen.M.L(1984). Optimal reinsurances. Scandinavian Actuarial,84(1), pp65-9.
37. M.J. Goovaerts, F. De Vylder, F. Mertens, R. Hardy(1980):An extension of an invariance property of Swiss premium calculation principle, ASTIN Bulletin 11(2), pp145-153.
38. E. Kremer(1982), Rating of largest claims and ECOMOR-reinsurance treaties for large portfolios, ASTIN Bull. pp47-56.
39. Young V.R.(1999)Optimal insurance under Wang's premium principle. Insurance:Math. Econ.Vol25(2), pp109-122.
40. D.C.M.Dickson & H.R.Waters(1999)Ruin probabilities with compounding assets, Insurance:Math. Econ. Vol 25,pp49-62.
41. Gerber,H.U. Shiu, E.S.W. (1998) On the time value of ruin. North American Actuarial. Journal,2,pp48-78.
42. A.G. de Kok (2003) Ruin probabilities with compounding assets for discrete time finite horizon problems, independent period claim sizes and general premium structure, Insurance:Math. Econ. Vo133, pp645-658.
43. Harrison, J. M. (1977) Ruin problems with compounding assets. Stoch. Proc. Appl.5, pp67-79.
44. Kahn, P. (1961). Some Remarks on a Recent Paper by Borch.ASTIN Bulletin 1, pp263-72.
45. Rothschild, M. J. E. Stiglitz (1975) Equilibrium in Competitive Insurance Markets. Technical Report No.170, IMSSS Stanford University.
46. Rothschild M. och J. Stiglitz (1976), Equilibrium in Competitive Insurance Markets:An Essay on the Economics of Imperfect Information, Quarterly Journal of Economics 90, pp629-649.
47. Gajek, L.& Zagrodny, D. (2000). Insurer's optimal reinsurance strategies, Insurance:Math. Econom.,27, pp227-240.
48. Pratt,J.W.(1964), Risk Aversion in the Small and in the Large, Econometrica,32, pp122-137.
49. Yaari, M.E. (1987). The dual theory of choice under risk. Econometrica 55, pp95-115.
50. Yaari, M. E.(1969)Some Remarks on Measures of Risk Aversion and on Their Uses,". Journal of Economic Theory. Vol.1, pp315-329.
51. Hipp, C. Vogt, M. (2003) Optimal dynamic XL reinsurance. ASTIN Bulletin 33, pp193-207.
52. Schmidli, H. (2001) Optimal proportional reinsurance policies in a dynamic setting. Scandinavian Actuarial Journal, pp55-68.
53. Moore, K.S.& Young, V.R. (2003). Pricing equitylinked pure endowments via the principle of equivalent utility, Insur. Math. Econ.33, pp497-516.
54. Borch, K. (1963). Recent developments in economic theory and their application to insurance. ASTIN Bulletin 2(3), pp322-341.
55. Yuen K. C.Guo J. Y.Wu X. Y.(2002)On a correlated aggregate claims model with Poisson and Erlang risk processes. Insurance:Mathematics & Economics,31,pp205-214.
56. H.U. Gerber(1981), The Esscher premium principle:A criticism, comment, ASTIN Bulletin 12 (2), pp139-140.
57. Yusong Cao, Nianqing Wan. (2009) Optimal proportional reinsurance and investment based on HJB equation. Insurance:Mathematics and Economics.45, pp157-162.
58. Young, V.R. (1999). Optimal insurance under Wang's premium principle. Insurance:Mathematics and Economics.25, pp109-122.
59. Haezendonck,J.M.Goovaerts (1982).A new Premium calculation principle based on Orlicz norms. Insurance Math. Econom. Vol.1, pp41-53.
60. Goovaerts, Marc J.Rob Kaas,Roger J.A. Laeven & Qihe Tang. (2004).A comonotonic image of independence for additive risk measures. Insurance:Math.Econom.Vol.35, pp581-594.
61. Quiggin J.(1981)Risk perception and risk aversion among Australian Farmers. Australian. Journal of Agricultural Economics; 25, pp160-169.
62. Dennenberg, D. (1990).Premium calculation:why standard deviation should be replaced by absolute deviation. ASTIN Bulletin,20, pp181-190.
63. Goovaerts,M.J.van.Heerwaarden,A.E.Kaas,R.(1989)Optimal reinsurance in relation to ordering of risk. Insur. Math. Econ.8, pp11-17.
64. Centeno, M. de L. (1986) Measuring the effects of reinsurance by the adjustment coefficient. Insurance Math. Econom 5, pp169-182.
65. Hipp, C. Vogt, M. (2003) Optimal dynamic XL reinsurance. ASTIN Bulletin 33, pp193-207.
66. Ross, S. A. (1981):Some Stronger Measures of Risk Aversion in the Small and the Large with Applications. Econometrica,49, pp621-638.
67. Deprez,O.H.U.Gerber(1985).On convex principles of premium calculation. Insurance:Math. Econom,4:pp179-189.
68. Marek Kaluszka (2005).Optimal reinsurance under convex principles of premium calculation. Insurance:Math. Econom 36:pp375-398.
69. Samuelson, P. A. (1952). Probability, Utility, and the Independence Axiom. Econometrica 20, pp670-678.
70. Ammeter, H (1964). Note concerning the distribution function of the total Loss excluding the largest individual clatms. ASTIN Bulletin 3, pp132-143.
71. Berlinger, B. (1972). Largest claims reinsurance (LCR) A quick method to calculate LCR-risk rates. ASTIN Bulletin 10, pp54-58.
72. B.Berliner(1972). Correlations between excess-of-loss reinsurance covers and reinsurance of the n largest claims, ASTIN Bull.6 (3) pp260-275.
73. Gerber, H. U. Goovaerts, M. (1981):On the representation of additive principles of premium calculation. S.A.J. pp221-227.
74. Kremer, E. (1982):A characterization of the Esscher-transformation. ASTIN Bulletin, pp58-60.
75. Buhlmann, H.Gagliardi, B. Gerber, H.U. E. Straub. (1977) Some inequalities for stop-loss premiums. ASTIN Bulletin 9(1), pp75-83;
76. Gerber, H.U.(1982):On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums. Insurance Math. Econom 1(1),pp 13-18.
77. Maria de Lourdes Centeno (2002). Measuring the effects of reinsurance by the adjustment coefficient in the Sparre Anderson model Insurance Math. Econom 30. pp37-49.
78. Vernic, R. (1997) On the bivariate generalized Poisson distribution. ASTIN Bulletbl 27, pp23-31.
79. CAI, J. TAN, K. S.(2007). Optimal retention fora stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bull.,37(1), pp93-112.
80. CAI, J.Tan, K. S.Weng, C.& Zhang, Y.(2008). Optimal reinsurance under VaR and CTE risk measures. Insurance Math. Econom.43, pp185-196.
81. Centeno, M. L.(2004). Retention and Reinsurance Programmes, in Encyclopedia of Actuarial Science,3,pp1443-1452.
82. Centeno, M. L. Guerra, M. (2008). The optimal reinsurance strategy-the individual claim case,http://cemapre.iseg.utl.pt/archive/preprints/ORS.pdf
83. Michel Denuit, Catherine Vermandele(1998).Optimal reinsurance and stop-loss order. Insurance Math. Econom 22.pp 229-233.
84. Ladoucette, S.A. Teugels, J.L.(2006). Reinsurance of large claims. Journal of Computational and Applied Mathematics 186 (1), pp163-190.
85. Stigler, G. (1950). The Development of Utility Theory:Ⅰ;Ⅱ. Journal of Political Economy,58,pp307-327; pp373-396.
86. Zaks, Y. Frostig, E. Levikson B. (2006) Optimal pricing of a hetero-geneous portfolio for a given risk level. ASTIN Bulletin,36(1), pp161-185.
87. G.C. Taylor(1985). A heuristic review of some ruin theory results. ASTIN Bulletin 15, pp73-88.
88. Centeno, M.L. (1997), Excess of Loss Reinsurance and the Probability of Ruin in Finite Horizon. ASTIN Bulletin, Vol.27, No1.pp59-70.
89. D.Samson,H.Thomas.(1983):Reinsurance Decision Making and Expected Utility. The Journal of Risk and Insurance, Vol.50, No.2, pp 249-264.
90. Gajek, L. Zagrodny, D.(2004). Optimal reinsurance under general risk measures, Insurance Math.Econom.34.pp105-112.
91. Benktander, G.(1975):A Note on Optimal Reinsurance. ASTIN Bulletin, VIII, pp154-164.
92. Danny Samson A. Wirth (1990). Expected utility strategic decision models for general insurers. Omega, Vo.18, (3), pp259-267.
93. Chris Starmer(2000). Developments in Non-Expected Utility Theory. Journal of Economic Literature. Vol.ⅩⅩⅩⅧ (6). pp332-382.
94. Guerra, M. Centeno, M. L.(2008). Optimal reinsurance policy:The adjustment coefficient and the expected utility criteria, Insurance Math. Econom.,42, pp529-539.
95. Jun Jiang, Qihe Tang. (2008). Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims. Insurance Math. Econom.Vol 43.pp431-436.
96. Lampaert, I. Walhin, J.F. (2005). On the Optimality of Proportional Reinsurance. Scandinavian Actuarial Journal(3),pp 225-239.
97. E.Kremer. (1982) Rating of largest claims and ECOMOR reinsurance treaties for large portfolios, ASTIN Bull.13 (1).pp47-56.
98. E.Kremer,(1990).The asymptotic efficiency of largest claims reinsurance treaties, ASTIN Bull.20 (1).pp11-22.
99. Glineur, F. J.F. Walhin (2006), de Finetti's retention problem for proportional reinsurance revisited, Unpublished manuscript.
100. Hansjorg Albrecher. Jozef L. Teugels.(2008) On excess-of-loss reinsurance. Teor. Tmovir. ta Matem. Statyst.No79, pp5-19.
101. Pesonen, M.I. (1984). Optimal reinsurances. Scandinavian Actuarial Journal 84, pp65-90.
102. Tversky, A.& Kahneman, D. (1991). Loss Aversion in Riskless Choice: A Reference Dependent Model. Quarterly Journal of Economics 106, pp1039-1061.
103. Matthew Rabin R. H. Thaler. (2000). Risk Aversion. Journal of Economic Perspectives. Vol15. pp219-232.
104. Amos Tversky, Daniel Kahneman.(1991)Loss Aversion in Riskless Choice A Reference-Dependent Model. The Quarterly Journal of Economics, Vol.106, No.4. pp 1039-1061.
105. E. Jouini, W. Schachermayer N. Touzi (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance 18. pp269-292.
106. Wang, S. S. V. R. Young (1998), Ordering risks:Expected utility theory versus Yaari's dual theory of risk, Insurance Math. Econom,22 (2): pp145-161.
107. 王梓坤.(1965)随机过程论.科学出版社.
108. 邓永录,梁之舜.(1998)随机点过程及其应用.北京：科学出版社.
109. Gerber.H. U.(1979),成世学,严颖译.数学风险论导引.世界图书出版社.
110. 劳斯.何声武,谢盛荣 等译.(1997)随机过程.北京：中国统计出版社.
111. [荷]R.卡尔斯,M.胡法兹,J.达纳,M.狄尼特著,唐启鹤,胡太忠,成世学译(2005).现代精算风险理论.(北京)科学出版社.
112. 赵晓芹,刘再明.(2008).广义二元复合Poisson风险模型破产概率的估计.统计研究,Vol.25(5),pp94-97.
113. 王晶刚,刘再明,周永卫(2005).保险系统中一类双险种风险模型的破产概率.数学理论与应用, Vol.25(1),pp39-42.
114. 江五元,刘再明.(2009)带阈限分红策略的一类更新风险模型.应用数学学报.Vol.32(3).pp454-466.
115. 赵晓芹,王国宝,刘再明.(2004)广义二元复合非齐次Poisson风险模型的破产概率.长沙理工大学学报(自然科学版)Vol.1.(3-4),pp74-77.
116. 赵晓芹,刘再明.(2004)一类双险种风险过程的破产概率的估计.数学理论与应用,Vol.24(1).pp97-100.
117. 刘东海,彭丹,刘再明.(2009)相依索赔的二项风险模型的破产问题.高校应用数学学报.Vol24(3).pp259-265.
118. 李俊海,刘再明.(2006)双更新风险模型.应用数学学报.Vol.29(3).pp344-351.
119. 刘汉进(2008).具有专属保险特性的国内保险公司发展及其影响.金融与经济,第2期,pp65-67.
120.龚日朝,李凤军(2001).双Poisson风险模型下的破产概率.湘潭师范学院学报(自然科学版)Vol.23.pp55-57.