两类带分红风险模型破产理论的相关结果
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摘要
风险理论的研究起源于瑞典精算师Lundberg,至今已有百年历史。随着保险公司经营规模的日益扩大和经营环境的不断变化,经典风险模型在很大程度上已不能模拟现实的风险状况。De Finetti于1957年率先提出了保险风险模型中的分红问题。红利的引入为保险公司的决策和经营提供了更多有价值的依据,特别地,为保险公司设计带红利的险种提供了理论依据。带分红的风险模型已经成为现代精算界和数学界研究的热门问题。为了与现代保险公司的实际需要相符,现代精算理论的很多研究是集中在随机环境下进行的。最近几年,对于一类更符合实际的变保费率的风险模型的研究也逐渐兴起。在考虑变保费率时,一般把保费率看成是风险自留的函数。除保费之外,利率因素对风险模型也有极大影响,带利率风险模型的研究也越来越引起人们的关注。
本文在前人工作的基础上,首先考虑了带分红的连续时间复合二项风险模型,给出期望折罚函数的递推式,破产前瞬时盈余和破产赤字的矩以及破产时刻的Laplace变换;接着对带有马氏调控利率、变保费率的复合Pascal模型及分红问题进行了研究,给出破产概率满足的递推式及破产时刻、破产前瞬时盈余、破产赤字等精算量联合分布满足的积分方程。
Studies of the risk theory were originated from the Sweden actuary Lundberg one hundred years ago. With the development of financial market and the change of the business environment of insurance company, the classical risk model can’t describe the real risk profile to a large extent. Dividend strategies for insurance risk models were firstly proposed by De Finetti in a paper presented to the International Congress of Actuaries in New York in 1957. Obviously, the theories about dividend strategies are very valuable in the devising and managing of products with dividends. The model with paying dividends has become a hot issue in the field of the modern actuarial science and mathematics. Correspond to the modern insurance company's actual need, many research in modern actuarial theory are focused on the random environment. The study of the risk model with varying premiums rate which fits the fact more precisely is also arising. Considering the variable premium rate, the premium rate is generally regarded as the function of surplus. Besides the premium rate, risk models with interest also have been studied by many researchers.
In this paper, we firstly study the expected discounted penalty function, the moments of surplus before ruin and deficit at ruin, and the Laplace transform of ruin time in the continuous-time compound binomial risk model with paying dividends; then we consider the compound Pascal model with paying dividends and variable premium under stochastic interest rates, we study the ruin probability and the joint distributions of this model.
本文在前人工作的基础上,首先考虑了带分红的连续时间复合二项风险模型,给出期望折罚函数的递推式,破产前瞬时盈余和破产赤字的矩以及破产时刻的Laplace变换;接着对带有马氏调控利率、变保费率的复合Pascal模型及分红问题进行了研究,给出破产概率满足的递推式及破产时刻、破产前瞬时盈余、破产赤字等精算量联合分布满足的积分方程。
Studies of the risk theory were originated from the Sweden actuary Lundberg one hundred years ago. With the development of financial market and the change of the business environment of insurance company, the classical risk model can’t describe the real risk profile to a large extent. Dividend strategies for insurance risk models were firstly proposed by De Finetti in a paper presented to the International Congress of Actuaries in New York in 1957. Obviously, the theories about dividend strategies are very valuable in the devising and managing of products with dividends. The model with paying dividends has become a hot issue in the field of the modern actuarial science and mathematics. Correspond to the modern insurance company's actual need, many research in modern actuarial theory are focused on the random environment. The study of the risk model with varying premiums rate which fits the fact more precisely is also arising. Considering the variable premium rate, the premium rate is generally regarded as the function of surplus. Besides the premium rate, risk models with interest also have been studied by many researchers.
In this paper, we firstly study the expected discounted penalty function, the moments of surplus before ruin and deficit at ruin, and the Laplace transform of ruin time in the continuous-time compound binomial risk model with paying dividends; then we consider the compound Pascal model with paying dividends and variable premium under stochastic interest rates, we study the ruin probability and the joint distributions of this model.
引文
[1] Lundberg F I. Approxmerad Framst?lling av Scannolik-hetsfunktionen.Ⅱ.àterf?rs?kring av Kollektivrisker. Uppsala: Almqvist & Wiksell, 1903.
[2] Bohlmann G. Die Theorie des Mittleren Risikos in des Lebensve-sicherung. Transactions of the International Congress of Actuaries, 1909, 1: 593~683.
[3] Lundberg F. F?rs?kringsteknisk Riskutjamming. F. Englunds, A. B. Boktryckeri, Stockholm, 1926.
[4] Cramér H. On the mathematical theory of risk. Stockholm: Skandia Jubilee Volume, 1930.
[5] Cramér H. On some questions connected with mathematical risk. California: University of California Press, 1954.
[6] Cramér H. Collective risk theory. Stockholm: Skandia Jubilee Volume, 1955.
[7] Cramér H. Half a century with probability theory: some personal recollection. The Annals of Probability, 1976, 4: 509~547.
[8] Gerber H U.数学风险论导引(成世学,严颖译).北京:世界图书出版公司, 1997.
[9]成世学.破产论研究综述.数学进展, 2002, 10: 403~422
[10]张波,代金.经济环境下引入投资的古典风险模型的破产概率.经济数学, 2005, 22: 111~117
[11] Gerber H U, Shiu E S W. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 1997, 21: 129~137.
[12] Gerber H U, Landry B. On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Mathematics and Economics, 1998, 22: 263~276.
[13] Gerber H U, Goovaerts M J, Kass R. On the probability and severity of ruin. ASTIN Bulletin, 1987, 17: 151~163.
[14] Dufresne F, Gerber H U. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics, 1988, 7: 193~199.
[15] Dufresne F, Gerber H U. The probability of ruin for the inverse Gaussian and related processes. Insurance: Mathematics and Economics, 1993, 12: 9~22.
[16] Dufresne F, Gerber H U. Risk theory for the compound Poisson processes that is perturbed by diffusion. Insurance: Mathematics and Economics, 1991, 10: 51~59.
[17] Shixue Cheng, Gerber H U, Shiu E S W. Discounted probabilities and ruin theory in the compound binomial model. Insurance: Mathematics and Economics, 2000, 26: 239~250.
[18] Gerber H U, Shiu E S W. Option pricing by Esscher transforms. Transactions of the Society ofActuaries, 1994, 46: 99~140.
[19] Gerber H U, Shiu E S W. Martingale approach to pricing perpetual American options. ASTIN Bulletin, 1994, 24: 195~220.
[20] Gerber H U, Shiu E S W. Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 1996, 18: 183~218.
[21] Gerber H U, Shiu E S W. Pricing perpetual options for jump processes. North American Actuarial Journal, 1998, 2: 101~107.
[22] Gerber H U, Shiu E S W. From ruin theory to pricing reset guarantees and perpetual put options. Insurance: Mathematics and Economics, 1999, 24: 3~14.
[23] Gerber H U. Martingales in risk theory. Vereinigung Schweizerischer Versicherungsma Thematiker, 1973, 3: 205~216.
[24] Li Wei, Rong Wu. The joint distribution of several important actuarial diagnostics in the classical risk model. Insurance: Mathematics and Economics, 2002, 30: 451~462.
[25] Rong Wu, Guojing Wang, Li Wei. Joint distributions of some actuarial random vectors containing the time of ruin. Insurance: Mathematics and Economics, 2003, 33: 147~161.
[26] Geber H U. Mathematical fun with the compound binomial process. Astin Bulletin, 1988, 18: 161~168.
[27] Gerber H U. Error bounds for the compound Poisson approximation. Insurance: Mathematics and Economics, 1984, 3: 191~194.
[28] Willmot G E. Ruin probabilities in the compound binomial model. Insurance: Mathematics and Economics, 1993, 12: 133~142.
[29] Dickson, D C M. Some comments on the compound binomial model. Astin Bulletin, 1994, 24:33~45
[30] Guoxin Liu, Jinyan Zhao. Joint distribution of some actuarial random vectors in the compound binomial model. Insurance: Mathematics and Economics, 2007, 40: 95~103.
[31] Sundt B, Teugels J L. Ruin estimates under interest force. Insurance: Mathematics and Economics, 1995, 16: 7~22.
[32] Hailiang Yang, Lihong Zhang. On the distribution of surplus immediately after ruin under interest force. Insurance: Mathematics and Economics, 2001, 29: 247~255.
[33] Hailiang Yang, Lihong Zhang. On the distribution of surplus immediately before ruin under interest force. Statistical Probability Letters, 2001, 55: 329~338.
[34]吴荣,杜勇宏.常利率下的更新风险模型.工程数学学报, 2002, 19(1): 46~54.
[35]林庆敏,汪荣明.带息力的更新风险模型下的破产概率的计算.华东师范大学学报, 2005,1: 46~52.
[36]李雅萍,尹清非.带利率的更新风险模型的破产概率的上界估计.数学理论与应用, 2005, 25(1): 106~108.
[37] Delbaen F, Haezendonck J. Classical risk theory in an economic environment. Insurance: Mathematics and Economics, 2000, 26: 239~250.
[38] Hélène Cossette, David Landriault,étienne Marceau. Compound binomial risk model in a markovian environment. Insurance: Mathematics and Economics, 2004, 35: 425~443.
[39]张茂军,南江霞.保费随机的复合二项风险模型的破产概率.科技通报, 2005, 21(3): 367~371.
[40]耿显民,万舒晨.随机利率下具有马氏调控的Pascal模型.应用数学学报, 2008, 31(4): 713~721.
[41]耿显民,汪颖.随机利率下马氏调控Pascal模型破产概率的估计.应用数学学报, 2010, 33(2):214~221.
[42] Petersen S. S. Calculation of ruin probabilities when the premium depends on the current reserve. Scandinavian Actuarial Journal, 1990, 147~159.
[43]蔡高玉.保费率为随机变量的风险模型和双复合Poisson风险模型, [硕士学位论文].南京:南京航空航天大学, 2004.
[44]耿显民,侯振挺.费率受制于位相半马氏过程的风险模型.南京航空航天大学学报, 2004, 36(5): 662~665.
[45] Helena Jasiulewicz. Probability of ruin with variable premium rate in a Markovian environment. Insurance: Mathematics and Economics, 2001, 29(2): 291-296.
[46]杨善朝.复合二项风险模型的破产概率的近似计算.广西师范大学学报, 2003, 21(4): 48~52.
[47] Cheng Shixue, Wu Biao. The survival probability in finite time period in fully discrete risk model. Journal of Chinese Universities, 1999, 14(B): 67~74.
[48]龚日朝,杨向群.复合二项风险模型下的生存概率.湘潭矿业学院学报, 2000, 15(4): 82~87.
[49]刘国欣,侯振挺,邹捷中.逐段决定马尔可夫骨架过程.长沙:湖南科学技术出版社, 2000.
[50] Guoxin Liu, Ying Wang, Bei Zhang. Ruin probability in the continuous-time compound binomial model. Insurance: Mathematics and Economics, 2005, 36: 303-316.
[51] Guoxin Liu, Ying Wang. On the expected discounted penalty function for the continuous-time compound binomial risk model. Statistics and Probability Letters, 2008, 78: 2446-2455.
[52] De Finetti. Su un’impostazione alternative della teoria colletive del rischio. Transactions of the XV International Congress of Actuaries, 1957, 2: 433-443.
[53] Gerber H U. On the probability of ruin in the presence of a linear dividend barrier. Scand. Acturial J. 1981, 2: 105-115.
[54] Siegl T, Tichy R F. A process with stochastic claim frequency and a linear dividend barrier. Insurance: Mathematics and Economics, 1999, 24: 51-65.
[55] Albrecher H., Kainhofer R., Risk theory with a nonlinear dividend barrier. Computing, 2002, 68: 289-311.
[56] Frostig, E. The expected time to ruin in a risk process with constant barrier via martingales. Insurance: Mathematics and Economics, 2005, 37: 216-228.
[57] Li, S. M., Lu, Y., Some optimal dividend problems in a Markov-modulated risk model. Working Paper. 2006.
[58] Tan, Jiyang, Yang, Xiangqun, The compound binomial model with randomized decisions on paying dividends. Insurance: Mathematics and Economics, 2006, 39:1-18.
[59] Zhaoyang Lu, Wei Xu, Decai Sun, et al. On the expected discounted penalty functions for two classes of risk processes under a threshold dividend strategy. Journal of Computational and Applied Mathematics, 2009, 232: 582~593
[60] Jun Cai, David C M Dickson, Ruin probabilities with a Markov chain interest model. Insurance: Mathematics and Economics, 2004, 35: 513~525.
[61] Bara Kim, Hwa-Sung Kim, Jeongsim Kim, A risk model with paying dividends and random environment. Insurance: Mathematics and Economics, 2008, 42: 717~726.
[62]王莹.带分红的复合帕斯卡模型及引文网模型的相关结果, [硕士学位论文].南京:南京航空航天大学, 2010.
[63]汪颖.随机环境下风险模型破产概率及复杂网络中的随机过程, [硕士学位论文].南京:南京航空航天大学, 2010.
[64]万舒晨.复合帕斯卡模型及其随机环境下相关问题的研究, [硕士学位论文].南京:南京航空航天大学, 2007.
[65]龚光鲁,钱敏平.应用随机过程教程.北京:清华大学出版社, 2004.
[2] Bohlmann G. Die Theorie des Mittleren Risikos in des Lebensve-sicherung. Transactions of the International Congress of Actuaries, 1909, 1: 593~683.
[3] Lundberg F. F?rs?kringsteknisk Riskutjamming. F. Englunds, A. B. Boktryckeri, Stockholm, 1926.
[4] Cramér H. On the mathematical theory of risk. Stockholm: Skandia Jubilee Volume, 1930.
[5] Cramér H. On some questions connected with mathematical risk. California: University of California Press, 1954.
[6] Cramér H. Collective risk theory. Stockholm: Skandia Jubilee Volume, 1955.
[7] Cramér H. Half a century with probability theory: some personal recollection. The Annals of Probability, 1976, 4: 509~547.
[8] Gerber H U.数学风险论导引(成世学,严颖译).北京:世界图书出版公司, 1997.
[9]成世学.破产论研究综述.数学进展, 2002, 10: 403~422
[10]张波,代金.经济环境下引入投资的古典风险模型的破产概率.经济数学, 2005, 22: 111~117
[11] Gerber H U, Shiu E S W. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 1997, 21: 129~137.
[12] Gerber H U, Landry B. On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Mathematics and Economics, 1998, 22: 263~276.
[13] Gerber H U, Goovaerts M J, Kass R. On the probability and severity of ruin. ASTIN Bulletin, 1987, 17: 151~163.
[14] Dufresne F, Gerber H U. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics, 1988, 7: 193~199.
[15] Dufresne F, Gerber H U. The probability of ruin for the inverse Gaussian and related processes. Insurance: Mathematics and Economics, 1993, 12: 9~22.
[16] Dufresne F, Gerber H U. Risk theory for the compound Poisson processes that is perturbed by diffusion. Insurance: Mathematics and Economics, 1991, 10: 51~59.
[17] Shixue Cheng, Gerber H U, Shiu E S W. Discounted probabilities and ruin theory in the compound binomial model. Insurance: Mathematics and Economics, 2000, 26: 239~250.
[18] Gerber H U, Shiu E S W. Option pricing by Esscher transforms. Transactions of the Society ofActuaries, 1994, 46: 99~140.
[19] Gerber H U, Shiu E S W. Martingale approach to pricing perpetual American options. ASTIN Bulletin, 1994, 24: 195~220.
[20] Gerber H U, Shiu E S W. Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 1996, 18: 183~218.
[21] Gerber H U, Shiu E S W. Pricing perpetual options for jump processes. North American Actuarial Journal, 1998, 2: 101~107.
[22] Gerber H U, Shiu E S W. From ruin theory to pricing reset guarantees and perpetual put options. Insurance: Mathematics and Economics, 1999, 24: 3~14.
[23] Gerber H U. Martingales in risk theory. Vereinigung Schweizerischer Versicherungsma Thematiker, 1973, 3: 205~216.
[24] Li Wei, Rong Wu. The joint distribution of several important actuarial diagnostics in the classical risk model. Insurance: Mathematics and Economics, 2002, 30: 451~462.
[25] Rong Wu, Guojing Wang, Li Wei. Joint distributions of some actuarial random vectors containing the time of ruin. Insurance: Mathematics and Economics, 2003, 33: 147~161.
[26] Geber H U. Mathematical fun with the compound binomial process. Astin Bulletin, 1988, 18: 161~168.
[27] Gerber H U. Error bounds for the compound Poisson approximation. Insurance: Mathematics and Economics, 1984, 3: 191~194.
[28] Willmot G E. Ruin probabilities in the compound binomial model. Insurance: Mathematics and Economics, 1993, 12: 133~142.
[29] Dickson, D C M. Some comments on the compound binomial model. Astin Bulletin, 1994, 24:33~45
[30] Guoxin Liu, Jinyan Zhao. Joint distribution of some actuarial random vectors in the compound binomial model. Insurance: Mathematics and Economics, 2007, 40: 95~103.
[31] Sundt B, Teugels J L. Ruin estimates under interest force. Insurance: Mathematics and Economics, 1995, 16: 7~22.
[32] Hailiang Yang, Lihong Zhang. On the distribution of surplus immediately after ruin under interest force. Insurance: Mathematics and Economics, 2001, 29: 247~255.
[33] Hailiang Yang, Lihong Zhang. On the distribution of surplus immediately before ruin under interest force. Statistical Probability Letters, 2001, 55: 329~338.
[34]吴荣,杜勇宏.常利率下的更新风险模型.工程数学学报, 2002, 19(1): 46~54.
[35]林庆敏,汪荣明.带息力的更新风险模型下的破产概率的计算.华东师范大学学报, 2005,1: 46~52.
[36]李雅萍,尹清非.带利率的更新风险模型的破产概率的上界估计.数学理论与应用, 2005, 25(1): 106~108.
[37] Delbaen F, Haezendonck J. Classical risk theory in an economic environment. Insurance: Mathematics and Economics, 2000, 26: 239~250.
[38] Hélène Cossette, David Landriault,étienne Marceau. Compound binomial risk model in a markovian environment. Insurance: Mathematics and Economics, 2004, 35: 425~443.
[39]张茂军,南江霞.保费随机的复合二项风险模型的破产概率.科技通报, 2005, 21(3): 367~371.
[40]耿显民,万舒晨.随机利率下具有马氏调控的Pascal模型.应用数学学报, 2008, 31(4): 713~721.
[41]耿显民,汪颖.随机利率下马氏调控Pascal模型破产概率的估计.应用数学学报, 2010, 33(2):214~221.
[42] Petersen S. S. Calculation of ruin probabilities when the premium depends on the current reserve. Scandinavian Actuarial Journal, 1990, 147~159.
[43]蔡高玉.保费率为随机变量的风险模型和双复合Poisson风险模型, [硕士学位论文].南京:南京航空航天大学, 2004.
[44]耿显民,侯振挺.费率受制于位相半马氏过程的风险模型.南京航空航天大学学报, 2004, 36(5): 662~665.
[45] Helena Jasiulewicz. Probability of ruin with variable premium rate in a Markovian environment. Insurance: Mathematics and Economics, 2001, 29(2): 291-296.
[46]杨善朝.复合二项风险模型的破产概率的近似计算.广西师范大学学报, 2003, 21(4): 48~52.
[47] Cheng Shixue, Wu Biao. The survival probability in finite time period in fully discrete risk model. Journal of Chinese Universities, 1999, 14(B): 67~74.
[48]龚日朝,杨向群.复合二项风险模型下的生存概率.湘潭矿业学院学报, 2000, 15(4): 82~87.
[49]刘国欣,侯振挺,邹捷中.逐段决定马尔可夫骨架过程.长沙:湖南科学技术出版社, 2000.
[50] Guoxin Liu, Ying Wang, Bei Zhang. Ruin probability in the continuous-time compound binomial model. Insurance: Mathematics and Economics, 2005, 36: 303-316.
[51] Guoxin Liu, Ying Wang. On the expected discounted penalty function for the continuous-time compound binomial risk model. Statistics and Probability Letters, 2008, 78: 2446-2455.
[52] De Finetti. Su un’impostazione alternative della teoria colletive del rischio. Transactions of the XV International Congress of Actuaries, 1957, 2: 433-443.
[53] Gerber H U. On the probability of ruin in the presence of a linear dividend barrier. Scand. Acturial J. 1981, 2: 105-115.
[54] Siegl T, Tichy R F. A process with stochastic claim frequency and a linear dividend barrier. Insurance: Mathematics and Economics, 1999, 24: 51-65.
[55] Albrecher H., Kainhofer R., Risk theory with a nonlinear dividend barrier. Computing, 2002, 68: 289-311.
[56] Frostig, E. The expected time to ruin in a risk process with constant barrier via martingales. Insurance: Mathematics and Economics, 2005, 37: 216-228.
[57] Li, S. M., Lu, Y., Some optimal dividend problems in a Markov-modulated risk model. Working Paper. 2006.
[58] Tan, Jiyang, Yang, Xiangqun, The compound binomial model with randomized decisions on paying dividends. Insurance: Mathematics and Economics, 2006, 39:1-18.
[59] Zhaoyang Lu, Wei Xu, Decai Sun, et al. On the expected discounted penalty functions for two classes of risk processes under a threshold dividend strategy. Journal of Computational and Applied Mathematics, 2009, 232: 582~593
[60] Jun Cai, David C M Dickson, Ruin probabilities with a Markov chain interest model. Insurance: Mathematics and Economics, 2004, 35: 513~525.
[61] Bara Kim, Hwa-Sung Kim, Jeongsim Kim, A risk model with paying dividends and random environment. Insurance: Mathematics and Economics, 2008, 42: 717~726.
[62]王莹.带分红的复合帕斯卡模型及引文网模型的相关结果, [硕士学位论文].南京:南京航空航天大学, 2010.
[63]汪颖.随机环境下风险模型破产概率及复杂网络中的随机过程, [硕士学位论文].南京:南京航空航天大学, 2010.
[64]万舒晨.复合帕斯卡模型及其随机环境下相关问题的研究, [硕士学位论文].南京:南京航空航天大学, 2007.
[65]龚光鲁,钱敏平.应用随机过程教程.北京:清华大学出版社, 2004.