两类含相关性的风险模型的研究
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摘要
这篇论文主要讨论了两类保费速率不为固定常数的含相关性的风险模型。第一类,保费速率由当前盈余决定。当盈余r不大于常数v时,保费速率为c(r)=c1+εr,反之,保费速率为c(r)=c2+εr,其中c1.c2,ε,r均为常数,本文给出了破产概率的表达式,并对索赔过程为指数分布的情形进行了具体计算；第二类,具有半马氏结构相依性的风险模型。该模型中,保费速率,索赔额分布,索赔间隔时间分布均被一离散时间马氏链决定。本文利用拉式变换对该模型的罚金折现函数进行分析,并给出了破产时刻、破产前盈余、破产赤字的任意阶矩的表达式。并以时间间隔分布满足Erlang(n)的模型为例进行了详细的计算。
This thesis studies two dependent risk models in which the premium rate is not a fixed constant. The first type is a time-dependent premium risk model in which premium rates are adjusted continuously according to the current level of an insurer's surplus. When surplus r(?)υ, the premium rate c(r)=c1+εr and when surplus r>υ, the premium rate c(r)=C2+εr, where c1, c2,εand r are all constants. We obtain expression for the ruin probability. For the exponential claim sizes, we exactly solve the ruin probability step-by-step. the second type is a Markov-dependent risk model in which the premium rate, the claim amounts and the interclaim time are depended by an irreducible discrete-time Markov chain. Based on the analysis of the discounted penalty function by means of Laplace-Stieltjes transforms, we drive moments of three characteristics of the ruin process. A renewal model with generalized Erlang(n)-interclaim times is contained as a special case.
This thesis studies two dependent risk models in which the premium rate is not a fixed constant. The first type is a time-dependent premium risk model in which premium rates are adjusted continuously according to the current level of an insurer's surplus. When surplus r(?)υ, the premium rate c(r)=c1+εr and when surplus r>υ, the premium rate c(r)=C2+εr, where c1, c2,εand r are all constants. We obtain expression for the ruin probability. For the exponential claim sizes, we exactly solve the ruin probability step-by-step. the second type is a Markov-dependent risk model in which the premium rate, the claim amounts and the interclaim time are depended by an irreducible discrete-time Markov chain. Based on the analysis of the discounted penalty function by means of Laplace-Stieltjes transforms, we drive moments of three characteristics of the ruin process. A renewal model with generalized Erlang(n)-interclaim times is contained as a special case.
引文
[1]Adan I., Kulkarni V., Single-server queue with Markov dependent interarrival and service times, Queueing Systems,45,113-134 (2003).
[2]Albrecher H., Asmussen S., Ruin probabilities and aggregate claims distributions for shot noise Cox process, Scandinavian Actuarial Journal,2,86-110 (2006).
[3]Albrechcr H., Boxma O., A ruin model with dependence between claim sizes and claim intervals, Insurance:Mathematics and Economic,35,245-254 (2004).
[4]Albrechcr H., Boxma O., On the discounted penalty function in a Markovdependent risk model, Insurance:Mathematics and Economics,37,650-672 (2005).
[5]Albrecher H.,Teugels J., Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability,43,257-273 (2006).
[6]Asmussen S., Ruin Probabilities, World Scientific, Singapore (2000).
[7]Boudreault M., Cossette H., Landrault D. and Marceau E., On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal,5,265-287 (2006).
[8]Boxma O., Perry D., A queueing model with dependence between service and interarrival times, European Journal of Operational Research,128 (3),611-624 (2001).
[9]Cheng Y., Tang Q., Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process, North American Actuarial Journal,7,1-12 (2003).
[10]Cossette H., Marceau E., The discrete-time risk model with correlated classes of business, Insurance:Mathematics and Economics,26,133-149 (2000).
[11]Cossette H., Marceau E. and Marri F., On the compound Poisson risk model with depen-dence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance:Mathematics and Economics,16,444-455 (2009).
[12]Cramer H., Collective Risk Theory, Stockholm:Skandia Jubilee Volume (1955).
[13]Dickson D., Hipp C., On the time to ruin for Erlang(2) risk process, Insurance:Mathe-matics and Economics,29,333-344 (2001).
[14]Dickson D., On the distribution of the surplus prior to ruin, Insurance Mathematics and Economics,11(3),191-207 (1992).
[15]Dickson D., Discussion on "On the time value of ruin" by H. Gerber and E. Shiu, North American Actuarial Journal,2(1),74 (1998).
[16]Dickson D., Drekic, S., The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models. Insurance Mathematics and Economics,34,97-107 (2004).
[17]Dufresne F., Gerber H.U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance Mathematics and Economics,7(3),193-199 (1988).
[18]Dufresne F., Gerber H.U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance:Mathematics and Economics,10,51-59 (1991).
[19]Eric C., David L., Gordon W.and Jae-Kyung W., Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insurance:Mathematics and Economics, 46,117-126(2010).
[20]Feller W., An Introduction to Probability Theory and its Application. Vol.Ⅱ, New York: John Wiley &Sons (1971)
[21]Gerber H.U., Martingale in risk theory, Mitt.Schweiz.Vers.Math.,73,205-216 (1973)
[22]Gerber H.U., Shiu W., On the time value of ruin, INorth American Actuarial Journal, 2(1),48-72 (1998)
[23]Gerber H.U., Shiu W., The joint distribution of the time of ruin, the surplus immediately before ruin and the deficit at ruin, Insurance:Mathematics and Economics,129-137 (1997).
[24]Gerber H.U., Shiu W., The time value of ruin in a Sparre Andersen Model, North American Actuarial Journal,9(2),49-69 (2005).
[25]Hu Y., Zhang Z.M. and Lan C.M., Ruin problems in a discrete Markov risk model, Statistics and Probability Letters,79,21-28 (2009).
[26]Peng J.Y.,Jin H., Ruin probability in a one-sided linear model with constant interest rate, Statistics and Probability Letters,80,662-669 (2010).
[27]Li S., Garrido J., On ruin for the Erlang(n) risk process, Insurance:Mathematics and Economics,34,391-408 (2004).
[28]Li S., Garrido J., On a general class of renewal risk process:analysis of the Gerber-Shiu function, Advances in Applied Probability, in press (2005).
[29]Lundberg I., Approximerad Framstallning av. Sannolikhetsfunktionen. Ⅱ, Atersforsakring av Kollektivrisker. Uppsala:Almqvist &Wiksell (1903).
[30]Marcus M., Minc H., A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston (1964).
[31]Meng Q.B., Zhang X. and Guo J.Y., On a risk model with dependence between claim sizes and claim interva, Statistics and Probability Letter,78,1727-1734 (2008).
[32]Zhou M., Cai J., A perturbed risk model with dependence between premium rates and claim sizes, Insurance:Mathematics and Economics,45,382-392 (2009).
[33]Zhang Z.M., Hu Y., On a risk model with stochastic premiums income and dependencebe-tween income and loss, Journal of Computational and Applied Mathematics,234,44-57 (2009).
[34]成世学,破产论研究综述,数学进展,31(5),403-422(2002).
[35]张连增，精算学中的随机过程,高等教育出版社(2006).
[2]Albrecher H., Asmussen S., Ruin probabilities and aggregate claims distributions for shot noise Cox process, Scandinavian Actuarial Journal,2,86-110 (2006).
[3]Albrechcr H., Boxma O., A ruin model with dependence between claim sizes and claim intervals, Insurance:Mathematics and Economic,35,245-254 (2004).
[4]Albrechcr H., Boxma O., On the discounted penalty function in a Markovdependent risk model, Insurance:Mathematics and Economics,37,650-672 (2005).
[5]Albrecher H.,Teugels J., Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability,43,257-273 (2006).
[6]Asmussen S., Ruin Probabilities, World Scientific, Singapore (2000).
[7]Boudreault M., Cossette H., Landrault D. and Marceau E., On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal,5,265-287 (2006).
[8]Boxma O., Perry D., A queueing model with dependence between service and interarrival times, European Journal of Operational Research,128 (3),611-624 (2001).
[9]Cheng Y., Tang Q., Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process, North American Actuarial Journal,7,1-12 (2003).
[10]Cossette H., Marceau E., The discrete-time risk model with correlated classes of business, Insurance:Mathematics and Economics,26,133-149 (2000).
[11]Cossette H., Marceau E. and Marri F., On the compound Poisson risk model with depen-dence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance:Mathematics and Economics,16,444-455 (2009).
[12]Cramer H., Collective Risk Theory, Stockholm:Skandia Jubilee Volume (1955).
[13]Dickson D., Hipp C., On the time to ruin for Erlang(2) risk process, Insurance:Mathe-matics and Economics,29,333-344 (2001).
[14]Dickson D., On the distribution of the surplus prior to ruin, Insurance Mathematics and Economics,11(3),191-207 (1992).
[15]Dickson D., Discussion on "On the time value of ruin" by H. Gerber and E. Shiu, North American Actuarial Journal,2(1),74 (1998).
[16]Dickson D., Drekic, S., The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models. Insurance Mathematics and Economics,34,97-107 (2004).
[17]Dufresne F., Gerber H.U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance Mathematics and Economics,7(3),193-199 (1988).
[18]Dufresne F., Gerber H.U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance:Mathematics and Economics,10,51-59 (1991).
[19]Eric C., David L., Gordon W.and Jae-Kyung W., Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insurance:Mathematics and Economics, 46,117-126(2010).
[20]Feller W., An Introduction to Probability Theory and its Application. Vol.Ⅱ, New York: John Wiley &Sons (1971)
[21]Gerber H.U., Martingale in risk theory, Mitt.Schweiz.Vers.Math.,73,205-216 (1973)
[22]Gerber H.U., Shiu W., On the time value of ruin, INorth American Actuarial Journal, 2(1),48-72 (1998)
[23]Gerber H.U., Shiu W., The joint distribution of the time of ruin, the surplus immediately before ruin and the deficit at ruin, Insurance:Mathematics and Economics,129-137 (1997).
[24]Gerber H.U., Shiu W., The time value of ruin in a Sparre Andersen Model, North American Actuarial Journal,9(2),49-69 (2005).
[25]Hu Y., Zhang Z.M. and Lan C.M., Ruin problems in a discrete Markov risk model, Statistics and Probability Letters,79,21-28 (2009).
[26]Peng J.Y.,Jin H., Ruin probability in a one-sided linear model with constant interest rate, Statistics and Probability Letters,80,662-669 (2010).
[27]Li S., Garrido J., On ruin for the Erlang(n) risk process, Insurance:Mathematics and Economics,34,391-408 (2004).
[28]Li S., Garrido J., On a general class of renewal risk process:analysis of the Gerber-Shiu function, Advances in Applied Probability, in press (2005).
[29]Lundberg I., Approximerad Framstallning av. Sannolikhetsfunktionen. Ⅱ, Atersforsakring av Kollektivrisker. Uppsala:Almqvist &Wiksell (1903).
[30]Marcus M., Minc H., A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston (1964).
[31]Meng Q.B., Zhang X. and Guo J.Y., On a risk model with dependence between claim sizes and claim interva, Statistics and Probability Letter,78,1727-1734 (2008).
[32]Zhou M., Cai J., A perturbed risk model with dependence between premium rates and claim sizes, Insurance:Mathematics and Economics,45,382-392 (2009).
[33]Zhang Z.M., Hu Y., On a risk model with stochastic premiums income and dependencebe-tween income and loss, Journal of Computational and Applied Mathematics,234,44-57 (2009).
[34]成世学,破产论研究综述,数学进展,31(5),403-422(2002).
[35]张连增，精算学中的随机过程,高等教育出版社(2006).