Logistic模型下变利率的破产问题的研究
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摘要
风险理论是对保险业所面临的各种风险进行定量分析的理论,在保险理论和实践中具有重要的意义。而破产理论则是风险理论中的重要组成部分和热点,破产理论及相关问题的研究可以作为保险公司一个十分有用的早期风险的警示手段,.因此具有十分重要的理论意义和现实意义。
随着保险市场发展的逐步成熟,新产品的不断开发,保险公司竞争日益激烈,经典的破产模型已经不能够满足现代保险业的需要,应对实际情况，许多作者对经典的破产模型进行了一系列改进。经典的风险模型没有考虑到利率的影响,在实际操作中,保险公司的大部分盈余来自于投资的收入一所以带利率的风险模型得到了广泛的研究。本文在前人研究的常利率风险模型的基础上,在假设利息力满足Logistic模型的前提下,提出变利率的风险模型,并且得出在此情况下变利率破产概率与常利率破产概率的关系。
Risk theory is the quantitative theory that analysis various risks that insurance com-panies confront.It is considerable important for insurance theory and practice, and the ruin theory is the important part and hotspot area in risk theory. As can supply a very usefull early-warning measure for the risk of the insurance company, the supply of the ruin theory and relative question has important theoretical and practical significance for the insurance company.
With the development of insurance market and the explosion of the new products, the competition of insurance company is more and more intense. The classical risk model has can not satisfy the requirement of the modern insurance. For answering readily the practice, more and more authors present a series of improvement to the classical risk model. In the classical risk theory, it is often assumed that there is no investment income. However, as we know, a large portion of the surplus of the insurance companies comes from investment income. So in resent years,the risk models with interest rate have been researched widely.In this article,on the base of the risk model with constant interest,we assumed the interest force satisfied Logistic model,and put forward the risk model with vary interest.With this hypothesis, we get the relationship between ruin probability in the vary interest model and that in the constant model.
随着保险市场发展的逐步成熟,新产品的不断开发,保险公司竞争日益激烈,经典的破产模型已经不能够满足现代保险业的需要,应对实际情况，许多作者对经典的破产模型进行了一系列改进。经典的风险模型没有考虑到利率的影响,在实际操作中,保险公司的大部分盈余来自于投资的收入一所以带利率的风险模型得到了广泛的研究。本文在前人研究的常利率风险模型的基础上,在假设利息力满足Logistic模型的前提下,提出变利率的风险模型,并且得出在此情况下变利率破产概率与常利率破产概率的关系。
Risk theory is the quantitative theory that analysis various risks that insurance com-panies confront.It is considerable important for insurance theory and practice, and the ruin theory is the important part and hotspot area in risk theory. As can supply a very usefull early-warning measure for the risk of the insurance company, the supply of the ruin theory and relative question has important theoretical and practical significance for the insurance company.
With the development of insurance market and the explosion of the new products, the competition of insurance company is more and more intense. The classical risk model has can not satisfy the requirement of the modern insurance. For answering readily the practice, more and more authors present a series of improvement to the classical risk model. In the classical risk theory, it is often assumed that there is no investment income. However, as we know, a large portion of the surplus of the insurance companies comes from investment income. So in resent years,the risk models with interest rate have been researched widely.In this article,on the base of the risk model with constant interest,we assumed the interest force satisfied Logistic model,and put forward the risk model with vary interest.With this hypothesis, we get the relationship between ruin probability in the vary interest model and that in the constant model.
引文
[1]Grandell J.Aspect of risk theory[M].New York:Springer Verlag.1991.
[2]Gerber H.U.数学风险论导引[M].成世学,严颖译.北京：世界图书出版公司,1979.
[3]成世学.破产论研究综述[J].数学进展.2002,31(5)：403-422.
[4]Lundberg.F.I.Approximerad Framstallning av Scannolikhetsfunktionen.Ⅱ,Aters-foraskring av Kollektivrisker.Uppasla:Almqvist& Wikset.1903.
[5]Cramer H.On the Mathods of Theory of Theory of Risk.Stoockholm:Skandia Jubilee Volume,1930.
[6]Cramer H.Mathematical Methods of Statistics.Princeton:Skandia and Princeton Univ.Press,Almqvist&Wiksell.1945.
[7]Cramer H.On some question connected with mathematical risk.University of Cali-fornia Publ.Statistics.1954,2:99-123.
[8]Cramer H.Collective Risk Theory. Stockholm:Skandia Jubilee Volume.1955.
[9]Feller W.An Introduction to Probability Theory and its Application[J].2nd Edi-tion.New York:John W iley&Sons.1971.
[10]Gerber H U.Martingale in risk theory.Mitt.Ver.Schweiz.Vers.Math.,1973,73:205-216.
[11]Ammeter H.A generalization of the collective theory of risk in regard to flunctuating basic probabilities[J].Skandinavisk Actuarial Journal.1948,171-198.
[12]Grandell J..Mixed poisson process[M].London:Chapman&hall,1997.
[13]Dickson D.C.Ruin probabilities for Erlang(2) risk process[J].Insurance:Mathematics and Economics.1998,22:251-262.
[14]Diskon D.C.On the time to ruin for Erlang(2)risk process[J].Insurance:Mathematics and Economics.2001,29:333-344.
[15]Shiu E S W.The probability of eventual ruin in the compound binomial mode[J].ASTTIN Bulletion.1989,19:179-190.
[16]Gerber H U.Mathematical fun with the compound binomialprocess[J].ASTIN Bulletin.1988,18:161-168.
[17]Cheng Shixue,Wu Biao.The survival probability in finite time period n fully discrete risk model.Applied Mathematics-AJournal of Chinese Universities[J].1999,14(B):67-74.
[18]成世学,伍彪.完全离散的经典风险模型[J].运筹学学报.1998,2(3)：42-54.
[19]成世学,朱仁栋.完全离散经典风险模型中的渐进解和Lundberg型不等式[J].高校应用数学学报A辑.2001,16(3)：348-358.
[20]Embrechts P,Kliippelbery C,Mikosch T.Modeling Extremal Events for Insurance and Finance[M].New York:Spring Verlag.1997.
[21]Mikosch,T.Heavy-tailed modeling in insurance. Stochastic Models[J].1997,13(4):799-816.
[22]Kluppelbery C.&Mikosch T.Large deviations of heavy-tailed randon sums with ap-plications in insurance and finance[J].J.Appl.Pro.1997,34:293-308.
[23]anxing Wang,Dafan Fang,Maoning Tang. Ruin Probabilities under a Markovian Risk Model[J].Acta Mathematics Applicate Sinica,English Series.2003,19(4):621-630.
[24]Sunt,B.,Teugels,J.L..Ruin estimates under interest force.nsurance:Mathematics and Economics.1995,16:7-22.
[25]Sunt,B.,Teugels,J.L..The adjustment function in ruin estimates under interest force[J].Insurance:Mathematics and conomics..1997,19:85-94.
[26]Yang HL,Zhang L.H.On the distribution of surplus immediately after ruin under interest force[J].Insurance:Mathematics and Economics.2001,29:247-255.
[2.7]Yang HL,Zhang L.H.The joint distribution of surplus immediately before ruin and the deficit at ruin under interest force[J].North American Actuarial Journal.2001,5(3):92-103.
[28]Kalashnikov,v.,Konstantinides,D..Ruin under interest force and subexponential claims:a simple treatment.Insurance:Mathematics and Economics.2000,27:145-149.
[29]Rongming Wang,Hailiang Yang,Haxing Wang. On the distribution of surplus immediately after,ruin under interest force and subexponential claims.Insurance:Mathematics and Economics.2004,35:703-714.
[30]吴荣,杜勇宏.常利率下的更新风险模型[J].工程数学学报.2002,19(1)：46-54.
[31]Stuart A.Klugman,Harry H.Panjer,Gordon E. Willmot损失模型：从数据到决策[M].吴岚译.北京：人民邮电出版社.2009.
[32]刘占国.利息理论[M].南开大学出版社.2000.
[33]Delbaen,F.,J.Haezendonck.Classical risk theory in an economic environment.Insur-ance:Mathematics and Economics.1987,6:85-116.
[2]Gerber H.U.数学风险论导引[M].成世学,严颖译.北京：世界图书出版公司,1979.
[3]成世学.破产论研究综述[J].数学进展.2002,31(5)：403-422.
[4]Lundberg.F.I.Approximerad Framstallning av Scannolikhetsfunktionen.Ⅱ,Aters-foraskring av Kollektivrisker.Uppasla:Almqvist& Wikset.1903.
[5]Cramer H.On the Mathods of Theory of Theory of Risk.Stoockholm:Skandia Jubilee Volume,1930.
[6]Cramer H.Mathematical Methods of Statistics.Princeton:Skandia and Princeton Univ.Press,Almqvist&Wiksell.1945.
[7]Cramer H.On some question connected with mathematical risk.University of Cali-fornia Publ.Statistics.1954,2:99-123.
[8]Cramer H.Collective Risk Theory. Stockholm:Skandia Jubilee Volume.1955.
[9]Feller W.An Introduction to Probability Theory and its Application[J].2nd Edi-tion.New York:John W iley&Sons.1971.
[10]Gerber H U.Martingale in risk theory.Mitt.Ver.Schweiz.Vers.Math.,1973,73:205-216.
[11]Ammeter H.A generalization of the collective theory of risk in regard to flunctuating basic probabilities[J].Skandinavisk Actuarial Journal.1948,171-198.
[12]Grandell J..Mixed poisson process[M].London:Chapman&hall,1997.
[13]Dickson D.C.Ruin probabilities for Erlang(2) risk process[J].Insurance:Mathematics and Economics.1998,22:251-262.
[14]Diskon D.C.On the time to ruin for Erlang(2)risk process[J].Insurance:Mathematics and Economics.2001,29:333-344.
[15]Shiu E S W.The probability of eventual ruin in the compound binomial mode[J].ASTTIN Bulletion.1989,19:179-190.
[16]Gerber H U.Mathematical fun with the compound binomialprocess[J].ASTIN Bulletin.1988,18:161-168.
[17]Cheng Shixue,Wu Biao.The survival probability in finite time period n fully discrete risk model.Applied Mathematics-AJournal of Chinese Universities[J].1999,14(B):67-74.
[18]成世学,伍彪.完全离散的经典风险模型[J].运筹学学报.1998,2(3)：42-54.
[19]成世学,朱仁栋.完全离散经典风险模型中的渐进解和Lundberg型不等式[J].高校应用数学学报A辑.2001,16(3)：348-358.
[20]Embrechts P,Kliippelbery C,Mikosch T.Modeling Extremal Events for Insurance and Finance[M].New York:Spring Verlag.1997.
[21]Mikosch,T.Heavy-tailed modeling in insurance. Stochastic Models[J].1997,13(4):799-816.
[22]Kluppelbery C.&Mikosch T.Large deviations of heavy-tailed randon sums with ap-plications in insurance and finance[J].J.Appl.Pro.1997,34:293-308.
[23]anxing Wang,Dafan Fang,Maoning Tang. Ruin Probabilities under a Markovian Risk Model[J].Acta Mathematics Applicate Sinica,English Series.2003,19(4):621-630.
[24]Sunt,B.,Teugels,J.L..Ruin estimates under interest force.nsurance:Mathematics and Economics.1995,16:7-22.
[25]Sunt,B.,Teugels,J.L..The adjustment function in ruin estimates under interest force[J].Insurance:Mathematics and conomics..1997,19:85-94.
[26]Yang HL,Zhang L.H.On the distribution of surplus immediately after ruin under interest force[J].Insurance:Mathematics and Economics.2001,29:247-255.
[2.7]Yang HL,Zhang L.H.The joint distribution of surplus immediately before ruin and the deficit at ruin under interest force[J].North American Actuarial Journal.2001,5(3):92-103.
[28]Kalashnikov,v.,Konstantinides,D..Ruin under interest force and subexponential claims:a simple treatment.Insurance:Mathematics and Economics.2000,27:145-149.
[29]Rongming Wang,Hailiang Yang,Haxing Wang. On the distribution of surplus immediately after,ruin under interest force and subexponential claims.Insurance:Mathematics and Economics.2004,35:703-714.
[30]吴荣,杜勇宏.常利率下的更新风险模型[J].工程数学学报.2002,19(1)：46-54.
[31]Stuart A.Klugman,Harry H.Panjer,Gordon E. Willmot损失模型：从数据到决策[M].吴岚译.北京：人民邮电出版社.2009.
[32]刘占国.利息理论[M].南开大学出版社.2000.
[33]Delbaen,F.,J.Haezendonck.Classical risk theory in an economic environment.Insur-ance:Mathematics and Economics.1987,6:85-116.