带有变利息率的离散时间风险模型的破产问题
摘要
N.L.Bower等人在《Actuarial Mathematics》一书中专门讨论了离散时间的风险模型，该模型将单位时间内收取的保费视为常数，每一时期的理赔量视为独立同分布的随机变量．H.Yang(1999)对常利息率的离散时间风险模型利用鞅方法得到破产概率的非指数上界。孙立娟、顾岚(2002)则在具有常利息率的离散时间模型下，讨论了破产前盈余分布、破产持续时间分布的问题。由于保费收入的时间和利息率对盈余过程的影响，J.Cai(2002)将这两种因素考虑在内，讨论了利息率分别具有独立同分布和一阶自回归结构时，在两种广义离散时间风险模型下的破产概率的一些问题。本文则在这两种广义风险模型下，当利息率{r_n，n=1，2…)为i．i．d．及一阶自回归结构r_n=ar_(n-1)+W_n时得出破产前一时刻剩余分布和破产持续时间分布所满足的公式。其次，本文讨论了广义复合Poisson风险模型在保单收入过程为～Poisson过程、个体索赔为伽玛分布情形下，讨论了更一般的有限时间内的生存概率问题，得到了较为满意的表达式。
In ((Actuarial Mathematics)) N. L .Bower etc. specially have discussed the discrete time risk model in which the premium income of per unit time is regarded as a constant, the claim amount of each period time is regarded as independent and identically distributed random variable. H. Yang (1999) gets the non-exponential bounds for ruin probability with constant interest by use of martingale methods. Under the discrete time risk model with constant interest, SUN Li-juan and GU Lan (2002) have discussed some ruin problems which are about the distributions of the surplus immediately before ruin and the time of the severity of ruin. Because of the effects of the timing of payments and interest on the surplus process, J. Cai(2002) considers the two factors , then discusses the problems of the ruin probability under the two generalized discrete time risk models respectively when the rates of interest are independent and identically distributed random variables and have a dependent autoregressive structure. In this
paper when the rates of interest{rn,n = 1,2,...}are i.i.d.and have a dependent autoregressive structure rn = arn-1 +Wn, we get the formulas of the distributions of the surplus immediately before ruin and the time of the severity of ruin under two generalized discrete time risk models. Secondly in this paper we discuss the common survival probability in finite time period under the generalized compound Poisson risk model in which the premium income process is a Poisson process and in case of Gamma's claim amounts, then we get more satisfied expressions.
In ((Actuarial Mathematics)) N. L .Bower etc. specially have discussed the discrete time risk model in which the premium income of per unit time is regarded as a constant, the claim amount of each period time is regarded as independent and identically distributed random variable. H. Yang (1999) gets the non-exponential bounds for ruin probability with constant interest by use of martingale methods. Under the discrete time risk model with constant interest, SUN Li-juan and GU Lan (2002) have discussed some ruin problems which are about the distributions of the surplus immediately before ruin and the time of the severity of ruin. Because of the effects of the timing of payments and interest on the surplus process, J. Cai(2002) considers the two factors , then discusses the problems of the ruin probability under the two generalized discrete time risk models respectively when the rates of interest are independent and identically distributed random variables and have a dependent autoregressive structure. In this
paper when the rates of interest{rn,n = 1,2,...}are i.i.d.and have a dependent autoregressive structure rn = arn-1 +Wn, we get the formulas of the distributions of the surplus immediately before ruin and the time of the severity of ruin under two generalized discrete time risk models. Secondly in this paper we discuss the common survival probability in finite time period under the generalized compound Poisson risk model in which the premium income process is a Poisson process and in case of Gamma's claim amounts, then we get more satisfied expressions.
引文
[1] Bowers N L, Gerber H U, HAckman J C, Jones D A, Nesbitt C J. Actuarial Mathematics [M],Society of Actuaries [M].Itasca, IL.,1986.
[2] Cai J. Discrete time risk models under rates of interest [J]. Prob. Eng. Inf. Sci. 2002,16:312-323.
[3] Yang H. Non-exponential bounds for ruin probability with interest effect included, Scandinavian Actuarial Journal[J],1999, 1:66-79.
[4] 孙立娟、顾岚．离散时间保险风险模型的破产问题[J]．应用概率统计，2002，18(3)：293-299．
[5] Cai J. Ruin probabilities with dependent rates of interest, J. Appl. Prob.,39(2002),312-323.
[6] GERBER H U. An Introduction to Mathematical Risk Theory[M]. Philadelphia : S S Huebner Foundation Monograph Series8 ,University of Pennsylvania ,1979.
[7] GRANDELL J .Aspects of Risk Theory[M].New York:Springer-Verlag, 1991.
[8] JHON A , BEEKMAN. A series for infinite time ruin probabilities[J]. insurance: Mathematics and Economics, 1985,4:129-134.
[9] SUN Li-juan, GU Lan .Stochastic simulation and analysis of claims and ruin for an insurance company[J].Application of Statistics and Management, 1999,18(4):25-30.
[10] 邓永录，梁之舜．随机点过程及其应用[M]．北京：北京科学出版社，1998，31-33．
[11] N．L．鲍尔斯等著，郑韫瑜，余跃年译．风险理论．上海：上海科学技术出版社，2001。
[12] 龚日朝，广义复合Poisson模型下有限时间内的生存概率，数学季刊，2003，18(2)：134-139．
[13] YAN Shi-jian, Liu Xiu-fang. Measure and Probability[M].Beijing: Beijing Normal University Press, 1994.
[14] F. De Vyldrer and M. j. Goovaerts, Recusive calculation of finite time ruin probabilities ,Insurance:Math. And Econom.,7(1988),1-7.
[15] T. Rolski, H. Schmidli, V. Schmidt and J.Teugels, Stochastic Processes for Insurance and Finance, Wiley and Sons, New York, 1999.
[16] ASMUSSEN, S. (2000).Ruin Probabilities. Word Scientific, Singapore.
[17] CAO, J. AND WANG, Y. (1991). The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28,473-479.
[18] CHENG, S. GERBER, H. U. AND SHIU, E. S. W. (2000).Discounted Probabilities and ruin theory in the compound binomial model .Insurance Math. econom. 26,239-250.
[19] DEVYLDER, F.E. AND MARCEAU, E. (1996).Classical numerical ruin probabilities. Scand. Actuarial J. 1996,109-123.
[20] GRANDELL, J. (1991).Aspects of Risk Theory. Spring, New York .
[21] GRANDELL, J. (1997) Mixed Poisson processes .Chapman and Hall , London.
[22] KELLISON, S. (1991).The Theory of Interest, 2nd edn. IRWIN, Boston.
[23] ROLSKI, T.,SCHMIDLI, V. AND TEUGELS, J.L. (1999).Stochastic processes for Insurance and Finance. John Wiley, Chichester.
[24] Ross, s. (1996).Stochastic Processes ,2nd edn. Jhon wiley, New York.
[25] SUNDT, B. AND TEUGELS, J.L. (1995).Ruin estimates under interest force .Insurance Math. Econom. 16,7-22.
[26] SUNDT, B .AND TEUGELS,J. L. (1997). The adjustment function in ruin estimates under interest force. Insurance Math. Econom. 19, 85-94.
[27] WILLMOT, G. E. (1996). A non-exponential generalization of an inequality arising in queueing and insurance risk.J. Appl. Prob. 33 ,176-183.
[28] WILLMOT, G.E. AND LIN, X.S. (2001).Lundberg Approximations for
Compound distributions with Insurance Applications. Springer, New York.
[29] S．G．Kellison著，尚汉冀译．利息理论．上海：上海科学技术出版社，1995．
[30] Gerber H.U. and E. S.W. shiu, On the time value of the time of ruin ,North American Actuarial Journal, 2:1(1998),48-72.
[31] Gerber H.U. and E.S.W. shiu, The joint distribution of the time of ruin ,the surplus immediately before ruin ,and the deficit at ruin ,Insurance :Mathematics and Economics, 21(1997),129-137.
[32] Sun Li-juan and H. Yang, Ruin theory in a discrete time risk model with interest incomes, To appear.
[2] Cai J. Discrete time risk models under rates of interest [J]. Prob. Eng. Inf. Sci. 2002,16:312-323.
[3] Yang H. Non-exponential bounds for ruin probability with interest effect included, Scandinavian Actuarial Journal[J],1999, 1:66-79.
[4] 孙立娟、顾岚．离散时间保险风险模型的破产问题[J]．应用概率统计，2002，18(3)：293-299．
[5] Cai J. Ruin probabilities with dependent rates of interest, J. Appl. Prob.,39(2002),312-323.
[6] GERBER H U. An Introduction to Mathematical Risk Theory[M]. Philadelphia : S S Huebner Foundation Monograph Series8 ,University of Pennsylvania ,1979.
[7] GRANDELL J .Aspects of Risk Theory[M].New York:Springer-Verlag, 1991.
[8] JHON A , BEEKMAN. A series for infinite time ruin probabilities[J]. insurance: Mathematics and Economics, 1985,4:129-134.
[9] SUN Li-juan, GU Lan .Stochastic simulation and analysis of claims and ruin for an insurance company[J].Application of Statistics and Management, 1999,18(4):25-30.
[10] 邓永录，梁之舜．随机点过程及其应用[M]．北京：北京科学出版社，1998，31-33．
[11] N．L．鲍尔斯等著，郑韫瑜，余跃年译．风险理论．上海：上海科学技术出版社，2001。
[12] 龚日朝，广义复合Poisson模型下有限时间内的生存概率，数学季刊，2003，18(2)：134-139．
[13] YAN Shi-jian, Liu Xiu-fang. Measure and Probability[M].Beijing: Beijing Normal University Press, 1994.
[14] F. De Vyldrer and M. j. Goovaerts, Recusive calculation of finite time ruin probabilities ,Insurance:Math. And Econom.,7(1988),1-7.
[15] T. Rolski, H. Schmidli, V. Schmidt and J.Teugels, Stochastic Processes for Insurance and Finance, Wiley and Sons, New York, 1999.
[16] ASMUSSEN, S. (2000).Ruin Probabilities. Word Scientific, Singapore.
[17] CAO, J. AND WANG, Y. (1991). The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28,473-479.
[18] CHENG, S. GERBER, H. U. AND SHIU, E. S. W. (2000).Discounted Probabilities and ruin theory in the compound binomial model .Insurance Math. econom. 26,239-250.
[19] DEVYLDER, F.E. AND MARCEAU, E. (1996).Classical numerical ruin probabilities. Scand. Actuarial J. 1996,109-123.
[20] GRANDELL, J. (1991).Aspects of Risk Theory. Spring, New York .
[21] GRANDELL, J. (1997) Mixed Poisson processes .Chapman and Hall , London.
[22] KELLISON, S. (1991).The Theory of Interest, 2nd edn. IRWIN, Boston.
[23] ROLSKI, T.,SCHMIDLI, V. AND TEUGELS, J.L. (1999).Stochastic processes for Insurance and Finance. John Wiley, Chichester.
[24] Ross, s. (1996).Stochastic Processes ,2nd edn. Jhon wiley, New York.
[25] SUNDT, B. AND TEUGELS, J.L. (1995).Ruin estimates under interest force .Insurance Math. Econom. 16,7-22.
[26] SUNDT, B .AND TEUGELS,J. L. (1997). The adjustment function in ruin estimates under interest force. Insurance Math. Econom. 19, 85-94.
[27] WILLMOT, G. E. (1996). A non-exponential generalization of an inequality arising in queueing and insurance risk.J. Appl. Prob. 33 ,176-183.
[28] WILLMOT, G.E. AND LIN, X.S. (2001).Lundberg Approximations for
Compound distributions with Insurance Applications. Springer, New York.
[29] S．G．Kellison著，尚汉冀译．利息理论．上海：上海科学技术出版社，1995．
[30] Gerber H.U. and E. S.W. shiu, On the time value of the time of ruin ,North American Actuarial Journal, 2:1(1998),48-72.
[31] Gerber H.U. and E.S.W. shiu, The joint distribution of the time of ruin ,the surplus immediately before ruin ,and the deficit at ruin ,Insurance :Mathematics and Economics, 21(1997),129-137.
[32] Sun Li-juan and H. Yang, Ruin theory in a discrete time risk model with interest incomes, To appear.