风险模型中的Gerber-Shiu函数及相关问题
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摘要
本论文主要考虑了Gerber-Shiu平均折现罚函数,该函数由Gerber, H. U.和Shiu, E. S. W (North American Actuarial Journal,2, 1998)首先介绍和研究.风险模型中的主要问题,如破产概率问题,破产时间、破产时赤字、破产前瞬时盈余三者的联合分布等问题均可由研究该函数而得出.本文讨论了不同模型中的Gerber-Shiu平均折现罚函数,建立了相关的积分-微分方程和更新方程,并讨论了方程解的情况.
论文的内容安排如下：
第一章简要介绍了Lundberg-Cramer经典风险模型,引入了Gerber-Shiu平均折现罚函数,列举了有关该函数的一些重要结果,并简单介绍了可能在后续章节要用到的一些随机过程中的概念和性质.
第二章首先利用鞅的方法得到了Cox风险模型中破产概率所满足的一些性质,主要引用的是Grandell(1991)的一些结果.第二节考虑了Cox风险模型中的Gerber-Shiu平均折现罚函数,建立了该函数所满足的积分-微分方程,得出了两状态模型时索赔量分布函数属于Kn-类时破产时间函数的具体表达式.
第三章讨论了索赔时间间距为指数分布和Erlang(2)分布混合时的平均折现罚函数,得到了该函数的积分-微分方程和更新方程,及Laplace变换,并得到了索赔量分布是Phase-type分布和Pareto分布时破产概率的明确表达式和近似表达式.
第四章的第一节研究了带阈值分红策略的索赔时间间距为Erlang(n)分布时的风险模型,建立了Gerber-Shiu函数的积分-微分方程和更新方程,讨论了更新方程的解.第二节介绍了多层分红策略时的情形,简述了一些主要结果,并讨论了带利率时的广义Erlang(n)模型的情况.
第五章先引入连续型和离散型风险模型中平稳更新风险模型和普通更新风险模型的Gerber-Shiu函数的关系式,在第二节讨论了带阈值分红策略的一类延迟更新风险模型,得到了该函数在延迟更新风险模型与普通更新风险模型中相应函数的关系表达式,平稳模型作为这类延迟模型的特殊情形,也作了相应的讨论,因普通更新风险模型中的Gerber-Shiu函数已被许多作者所研究,并得到了一些较深刻的结果,因此,以普通更新风险模型为基础来研究延迟风险模型有其理论基础.
第六章讨论了带扩散干扰更新风险模型,当索赔时间间距是广义Erlang(n)分布时,考虑了破产前最大盈余分布函数,得到了该分布函数的积分-微分方程,并研究了其对应的齐次方程的解.最后,介绍了Gerber-Shiu函数在带阈值分红策略时所满足的一些性质.
The thesis mainly considers the Gerber-Shiu expected discounted penalty function introduced and studied by Gerber and Shiu [North American Actuarial Journal,2,1998]. The function includes many important problems in risk models, for example, ruin probability, the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, and so on. The thesis studies the Gerber-Shiu expected discounted penalty function in the different risk models, obtains the corresponding integro-differential equations and renewal equations, and discusses the solutions of these equations.
The thesis is structured as follows:
In Chapter 1, we simply introduce the Lundberg-Cramer classical risk model in section 1.1, and the Gerber-Shiu expected discounted penalty function in section 1.2. Furthermore, the basic knowledge needed in the thesis listed in section 1.3.
In Chapter 2, we show that the properties of ruin probability in the Cox risk model by Martingale approach in section 2.1, some results can be found in Grandell (1991). In section 2.2, the Gerber-Shiu function is examined in a class of Cox risk models, an integro-differential equation is derived. In the two-state model, explicit formulas for ruin time function are obtained when claim size distribution belongs to the Kn-family.
In Chapter 3, we consider a class of renewal risk process in which the claim inter-arrival times are the mixture of exponentials and Erlang(2) distributions. A integro-differential equation and renewal equation for the expected discounted penalty function are derived, and we also discuss the Laplace transform. We obtain the explict solution and the asymptotic one in the case where the individenal claim amount distribution is Phase-type and Pareto, respectively.
In Chapter 4, we consider a renewal risk process with a threshold dividend strategy in which the claim inter-arrival times are Erlang(n) distributed. Two integro-differential equations and a renewal one for the Gerber-Shiu function are obtained. We also discuss the solution of the renewal equation in section 4.1. Gerber-Shiu functions in generalized Erlang(n) risk model with multi-layer dividend strategy and with interest are considered in section 4.2.
In Chapter 5, the Gerber-Shiu functions are considered in the con-tinuous and decrete stationary renewal risk model, where they are expressed in terms of the same functions in the ordinary renewal risk model in section 5.1. In section 5.2, we consider a class of delayed renewal risk processes with a threshold dividend strategy, the main result is an expression of the Gerber-Shiu function in the delayed renewal risk model in terms of the corresponding function in the ordinary renewal risk model. We state that the stationary renewal risk model is the special case of the delayed renewal risk one in this section.
In Chapter 6, we study the distribution of the maximum surplus before ruin in a Sparre Andersen risk model with the inter-claim times being generalized Erlang(n) distributed perturbed by diffusion, two integro-differential equations are derived, the solution of the first corresponding homogenous integro-differential equation is considered in section 6.1. The Gerber-Shiu function is also considered in a Sparre Andersen risk model perturbed by diffusion in section 6.2.
论文的内容安排如下：
第一章简要介绍了Lundberg-Cramer经典风险模型,引入了Gerber-Shiu平均折现罚函数,列举了有关该函数的一些重要结果,并简单介绍了可能在后续章节要用到的一些随机过程中的概念和性质.
第二章首先利用鞅的方法得到了Cox风险模型中破产概率所满足的一些性质,主要引用的是Grandell(1991)的一些结果.第二节考虑了Cox风险模型中的Gerber-Shiu平均折现罚函数,建立了该函数所满足的积分-微分方程,得出了两状态模型时索赔量分布函数属于Kn-类时破产时间函数的具体表达式.
第三章讨论了索赔时间间距为指数分布和Erlang(2)分布混合时的平均折现罚函数,得到了该函数的积分-微分方程和更新方程,及Laplace变换,并得到了索赔量分布是Phase-type分布和Pareto分布时破产概率的明确表达式和近似表达式.
第四章的第一节研究了带阈值分红策略的索赔时间间距为Erlang(n)分布时的风险模型,建立了Gerber-Shiu函数的积分-微分方程和更新方程,讨论了更新方程的解.第二节介绍了多层分红策略时的情形,简述了一些主要结果,并讨论了带利率时的广义Erlang(n)模型的情况.
第五章先引入连续型和离散型风险模型中平稳更新风险模型和普通更新风险模型的Gerber-Shiu函数的关系式,在第二节讨论了带阈值分红策略的一类延迟更新风险模型,得到了该函数在延迟更新风险模型与普通更新风险模型中相应函数的关系表达式,平稳模型作为这类延迟模型的特殊情形,也作了相应的讨论,因普通更新风险模型中的Gerber-Shiu函数已被许多作者所研究,并得到了一些较深刻的结果,因此,以普通更新风险模型为基础来研究延迟风险模型有其理论基础.
第六章讨论了带扩散干扰更新风险模型,当索赔时间间距是广义Erlang(n)分布时,考虑了破产前最大盈余分布函数,得到了该分布函数的积分-微分方程,并研究了其对应的齐次方程的解.最后,介绍了Gerber-Shiu函数在带阈值分红策略时所满足的一些性质.
The thesis mainly considers the Gerber-Shiu expected discounted penalty function introduced and studied by Gerber and Shiu [North American Actuarial Journal,2,1998]. The function includes many important problems in risk models, for example, ruin probability, the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, and so on. The thesis studies the Gerber-Shiu expected discounted penalty function in the different risk models, obtains the corresponding integro-differential equations and renewal equations, and discusses the solutions of these equations.
The thesis is structured as follows:
In Chapter 1, we simply introduce the Lundberg-Cramer classical risk model in section 1.1, and the Gerber-Shiu expected discounted penalty function in section 1.2. Furthermore, the basic knowledge needed in the thesis listed in section 1.3.
In Chapter 2, we show that the properties of ruin probability in the Cox risk model by Martingale approach in section 2.1, some results can be found in Grandell (1991). In section 2.2, the Gerber-Shiu function is examined in a class of Cox risk models, an integro-differential equation is derived. In the two-state model, explicit formulas for ruin time function are obtained when claim size distribution belongs to the Kn-family.
In Chapter 3, we consider a class of renewal risk process in which the claim inter-arrival times are the mixture of exponentials and Erlang(2) distributions. A integro-differential equation and renewal equation for the expected discounted penalty function are derived, and we also discuss the Laplace transform. We obtain the explict solution and the asymptotic one in the case where the individenal claim amount distribution is Phase-type and Pareto, respectively.
In Chapter 4, we consider a renewal risk process with a threshold dividend strategy in which the claim inter-arrival times are Erlang(n) distributed. Two integro-differential equations and a renewal one for the Gerber-Shiu function are obtained. We also discuss the solution of the renewal equation in section 4.1. Gerber-Shiu functions in generalized Erlang(n) risk model with multi-layer dividend strategy and with interest are considered in section 4.2.
In Chapter 5, the Gerber-Shiu functions are considered in the con-tinuous and decrete stationary renewal risk model, where they are expressed in terms of the same functions in the ordinary renewal risk model in section 5.1. In section 5.2, we consider a class of delayed renewal risk processes with a threshold dividend strategy, the main result is an expression of the Gerber-Shiu function in the delayed renewal risk model in terms of the corresponding function in the ordinary renewal risk model. We state that the stationary renewal risk model is the special case of the delayed renewal risk one in this section.
In Chapter 6, we study the distribution of the maximum surplus before ruin in a Sparre Andersen risk model with the inter-claim times being generalized Erlang(n) distributed perturbed by diffusion, two integro-differential equations are derived, the solution of the first corresponding homogenous integro-differential equation is considered in section 6.1. The Gerber-Shiu function is also considered in a Sparre Andersen risk model perturbed by diffusion in section 6.2.
引文
[1]Cramer, H. Collective Risk Theory. Stockholm:Skandia Jubilee Volume.1955
[2]Gerber, H.U. An Introduction to Mathematical Risk Theory. Monograph Series 8, Huebner Foundation, Philadelphia.1979
[3]Grandell,J. Aspects of Risk Theory. Springer-Verlag,1991
[4]Rolski, T., Schmidli, H., Schmidet, V., et al. Stochastic Processes for Insurance and Finance. New York:John Wiley & Sons,1999
[5]Asmussen, S. Ruin Probabilities. Singapore:World Scientific,2000
[6]De Finetti B. Su un'impostazione alternativadella teoria collettival del rischio. Proceedings of the transactions of the XV international congress of actuaries, 1957,2:433-443
[7]Buhlmann, H. Mathematical Mothods in Risk Theory. New York:Springer-Verlag,1970
[8]Segerdahl, C. On some distribution in time-connected with the collective theory of risk. Scand. Actuarial J.1970:167-192
[9]Gerber, H.U. Martingales in Risk Theory. Mitteilungen der Schweizer Vere-inigung der Versicherungsmathematiker,1973,73:205-206
[10]Gerber, H.U. On the probability of ruin in the presence of a linear dividend barrier. Scand. Actuarial J.1981,2:105-115
[11]Paulsen, J., Gjessing, H. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance:Mathematics and Economics, 1997,20:215-223
[12]Albrecher, H., Kainhofer, R. Risk theory with a nonlinear dividend barrier. 2002,68:289-311
[13]Hojgaard, B. Optimal dynamic premium control in non-life insurance:maxim-izing dividend payouts. Scand. Actuarial J.2002,225-245
[14]Gerber, H.U. Der Einfluss von Zins auf die Ruinwahrscheinlichkeit. Mitteilun-gen der Schweizerischen Vereinigung der Versicherungsmathematiker,1971: 63-70
[15]Delbaen, F., Haezendonck, J. Classical risk theory in an economic environ-ment. Insurance:Mathematics and Economics,1987,6:85-116
[16]Sundt, B., Teugels, J.L. Ruin estimates under interest force. Insurance:Mathe-matics and Economics,1995,16:7-22
[17]Sundt, B., Teugels, J.L. The adjustment function in ruin estimates under inter-est force. Insurance:Mathe-matics and Economics,1997,19:85-94
[18]Cai, J., Dickson, D.C.M. On the expected discounted penalty function at ruin of a surplus process with interest. Insurance:Mathematics and Economics, 2002,30:389-404
[19]Wu, R., Wang, G, Zhang, C. On a joint distribution for the risk process with constant interest force. Insurance:Mathematics and Economics,2005,36: 365-374
[20]Andersen, E.S. On the collective theory of risk in case of contagion between claims. Bulletin of the Institute of Mathematics and its Applications.1957,12: 257-279
[21]Dickson, D.C.M. On a class of renewal risk process. North American Actuarial Journal,1998,2(3):60-68
[22]Dickson, D.C.M., Hipp, C. Ruin probabilities for Erlang(2) risk process. Insur-ance:Mathematics and Economics,1998,22:251-262
[23]Dickson, D.C.M., Hipp, C. On the of ruin for Erlang(2) risk processes. Insur-ance:Mathematics and Economics,2001,29:333-334
[24]Cheng, Y.B., Tang, Q.H. Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal,2003,7: 1-12
[25]Li, S., Garrido, J. On ruin for Erlang(n) risk process. Insurance:Mathematics and Economics,2004,34:391-408
[26]Willmot, G.E. On the discounted penalty function in the renewal risk model with general interclaim times. Insurance:Mathematics and Economics,2007, 41:17-31
[27]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:Mathe-matics and Economics,1997,21:129-137
[28]Gerber, H.U., Shiu, E.S.W. On the time value of ruin. North American Actu-arial Journal,1998,2:48-72
[29]Lin, X.S., Willmot, G.E. Analysis of a defective renewal equation arising in ruin theory. Insurance:Mathematics and Economics,1999,25:63-84
[30]Willmot,G.E., Dickson,D.C.M. The Gerber-Shiu discounted penalty function in the stationary renewal model. Insurance:Mathematics and Economics,2003, 32:403-411
[31]Pavlova, K.P., Willmot, G.E. The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance:Mathematics and Economics,2004,35:267-277
[32]Gerber, H.U., Shiu, E.S.W. The time value of ruin in a Sparre Andersen model. North American Actuarial Journal,2005,9(2):49-84
[33]Li, S., Lu, Y. On the expected discounted penalty functions for two classes of risk processes. Insurance:Mathematics and Economics,2005,36:179-193
[34]Ahn, S., Badescu, A.L. On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals. Insurance:Mathematics and Economics,2007,41:234-249
[35]Lin, X.S., Willmot, G.E., Drekic, S. The classical risk model with a constant dividend barrier:analysis of the Gerber-Shiu discounted penalty function. Insurance:Mathematics and Economics,2003,33:551-556
[36]Li, S., Garrido, J. On a class of renewal risk models with a constant dividend barrier. Insurance:Mathematics and Economics,2004,35:691-701
[37]Lin, X.S., Pavlova, K.P. The compound Poisson risk model with a threshold dividend strategy. Insurance:Mathematics and Economics,2006,38:57-80
[38]Yuen, K.C., Wang, G.J., Li, W.K. The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier. Insur-ance:Mathematics and Economics,2007,40:104-112
[39]Hu Yang, Zhimin Zhang. Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy. Insurance:Mathematics and Economics,2008,42:984-991
[40]Albrechar, H., Hartinger, J., Tichy, R. On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier. Scandinavian actuarial journal,2005,2:130-136
[41]Gerber H.U. An extension of the renewal equation and its application in the collective theory of risk.Scandinavian Actuarial journal,1970,205-210
[42]Sabine S. Ruin probabilities in perturbed risk models. Insurance:Mathematics and Economics,1998,22:93-104
[43]Chiu, S.N., Yin, C.C. The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance: Mathematics and Economics.2003,33:59-66
[44]Wang G.J., Wu R. A generalization of risk model perturbed by diffusion. Appl.Aath-Juc,1999,14(4):337-341
[45]Wang G. J., Wu R. Some distribution for classical risk process that is perturbed by diffusion.Insurance:Mathematics and Economics,2000,26:15-24
[46]Claudio, M., Giovanni L.T. Asymptotic results for perturbed risk processes with delayed claims. Insurance:Mathematics and Economics,2004,34:307-320
[47]Tsai, C.C.L., Willmot, G.E. On the moments of the surplus process perturbed by diffussion. Insurance:Mathematics and Economics,2002,31:327-350
[48]Gao, H., Yin, C.C. The perturbed Sparre Andersen model with a threshold dividend strategy. Journal of Computational and Applied Mathematics.2008, 220:394-408
[49]邓永录,梁之瞬.随机点过程及其应用.北京：科学出版社,1998
[50]钱敏平,龚光鲁.随机过程论.北京：北京大学出版社,1992
[51]侯振挺,刘万荣,刘再明等.马尔可夫骨架过程-混杂系统模型.湖南：湖南出版社,2000
[52]Asmussen, S. Applied Probability and Queues. John Wiley & Sons,1987
[53]Feller W. An introduction to probability theory and its application. Vol.Ⅱ. New York:John Wiley & Sons,1971
[54]Revuz, D.& Yor, M., Continuous Martingales and Brownian Motion. Sprin-ger, Berlin,1991
[55]何声武,汪嘉冈,严加安.半鞅与随机分析.北京.科学出版社,1995
[56]严加安.鞅与随机分析.上海：上海科学技术出版社,1981
[57]严加安.测度论讲义.北京：科学出版社,2000
[58]Applebaum, D. Lvey processes and stochastic calculus. Cambridge University Press,2004
[59]Ammeter, H. A generalization of the collective theory of risk in regard to fluctuating basic probabilities. Skand. Aktuar Tidskr,1948
[60]Reinhard, J.M. On a class of semi-Markov risk models obtained as classical risk models in a markovian environment. Astin Bulletin ⅩⅣ,1984
[61]Asmussen, S. Risk theory in markovian environment. Scand actuarial J.1989
[62]何树红,徐兴富.双Cox风险模型.云南大学学报,2004,26(4)：275-278
[63]刘艳,胡亦钧.马氏环境下带扰动的Cox相关风险模型破产概率的上界估计.数学年刊,2004,25A(5)：597-586
[64]Li, S., Lu, Y. On the probability of ruin in a Markov-modulated risk model. Insurance:Mathematics and Economics,2005,37:522-532
[65]Neuts, M.F. Algorithms for the waiting time distribution under various queue disciplines in the M/G/1 queue with service time distribution of phase type. Algorithmic Methods in Probability, TIMS Studies in the Management Sciences,1977
[66]Gerber,H.U., Shiu,E.S.W. Discussion of "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process". North American Actuarial Journal,2003(a),7(3):117-119
[67]Gerber,H.U., Shiu,E.S.W. "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process, "Yebin Cheng and Qihe Tang. North American Actuarial Journal,2003(b),7(4):96-101
[68]Shuangming, Li. Discussion of "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process". North American Actuarial Journal,2003,7(3):119-122
[69]Lin, X.S. Discussion of "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process". North American Actuarial Journal, 2003,7(3):122-124
[70]Tsai, C.C.L., Sun, L.J. On the discounted distribution function for the Erlang(2) risk process. Insurance:Mathematics and Economics,2004,35:5-19
[71]Zhao, X.H., Yin, C.C. Ruin problems for a Sparre Andersen risk model.应用概率统计,2005,21(4)：431-442
[72]李俊海.鞅在风险模型中的应用：[博士学位论文].长沙：中南大学,2007
[73]Embrechts, P., Goldie, C.M., Veraverbebe, N. Subexponentiality and infinite divisibility. Probability theory and related fields,1979,49:335-347
[74]Gerber, H.U., Shiu, E.S.W. Discussion of moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal,2003,17(4):96-101
[75]Lin, X.S., Sendova, K.P. The compound Poisson risk model with multiple thre-sholds. Insurance:Mathematics and Economics,2008,42:617-627
[76]Li, S., Garrido, J. The Gerber-Shiu function in a Sparre Andersen risk process perturbed by diffusion. Scandinavian Actuarial Journal,2005,161-186
[77]Xing, Y, Wu, R. The expected valued of a penalty function for a PDMP insur-ance risk model.应用概率统计,2005,21(4)：397-402
[78]Mikhlin, S.G. Integral equations. Pergamon Press, London,1957
[79]Embrechts, P., Kluppelberg, C., Mikosck, T. Modelling Extremal Events. Springer, Berlin,1997
[80]Willmot, G., Lin, X. Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York,2001
[81]Gerber, H.U. Mathematical fun with the compound binomial process. ASTIN Bulletin,1988,18:161-168
[82]Shiu, E.S.W. Probability of eventual ruin in the compound binomial model. ASTIN Bulletin,1988,19:179-190
[83]Willmot,G. E.. A note on a class of delayed renewal risk processes. Insurance: Mathematics and Economics,2004,34:251-257
[84]Zhenhua Bao, Zhongxing Ye. The Gerber-Shiu discounted penalty function in the delayed renewal risk process with random income. Applied Mathematics and Computation.2007,184:857-863
[85]Li, S., Dickson,D.C.M. The maximum surplus before ruin in an Erlang(n) risk process and related problems. Insurance:Mathematics and Economics,2006, 38:529-539
[86]Dufresne, F., Gerber, H.U. Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance:Mathematics and Economics,1991,10: 51-59
[87]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
[88]Tsai, C.C.L., Willmot, G.E. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance:Mathematics and Economics,2002,30:51-66
[89]Tsai, C.C.L. On the expectations of the present values of the time of ruin perturbed by diffusion. Insurance:Mathematics and Economics,2003,32: 413-429
[90]Sarkar, J., Sen, A. Weak convergence approach to compound Poisson risk pro-cesses perturbed by diffusion. Insurance:Mathematics and Economics,2005, 36:421-432
[2]Gerber, H.U. An Introduction to Mathematical Risk Theory. Monograph Series 8, Huebner Foundation, Philadelphia.1979
[3]Grandell,J. Aspects of Risk Theory. Springer-Verlag,1991
[4]Rolski, T., Schmidli, H., Schmidet, V., et al. Stochastic Processes for Insurance and Finance. New York:John Wiley & Sons,1999
[5]Asmussen, S. Ruin Probabilities. Singapore:World Scientific,2000
[6]De Finetti B. Su un'impostazione alternativadella teoria collettival del rischio. Proceedings of the transactions of the XV international congress of actuaries, 1957,2:433-443
[7]Buhlmann, H. Mathematical Mothods in Risk Theory. New York:Springer-Verlag,1970
[8]Segerdahl, C. On some distribution in time-connected with the collective theory of risk. Scand. Actuarial J.1970:167-192
[9]Gerber, H.U. Martingales in Risk Theory. Mitteilungen der Schweizer Vere-inigung der Versicherungsmathematiker,1973,73:205-206
[10]Gerber, H.U. On the probability of ruin in the presence of a linear dividend barrier. Scand. Actuarial J.1981,2:105-115
[11]Paulsen, J., Gjessing, H. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance:Mathematics and Economics, 1997,20:215-223
[12]Albrecher, H., Kainhofer, R. Risk theory with a nonlinear dividend barrier. 2002,68:289-311
[13]Hojgaard, B. Optimal dynamic premium control in non-life insurance:maxim-izing dividend payouts. Scand. Actuarial J.2002,225-245
[14]Gerber, H.U. Der Einfluss von Zins auf die Ruinwahrscheinlichkeit. Mitteilun-gen der Schweizerischen Vereinigung der Versicherungsmathematiker,1971: 63-70
[15]Delbaen, F., Haezendonck, J. Classical risk theory in an economic environ-ment. Insurance:Mathematics and Economics,1987,6:85-116
[16]Sundt, B., Teugels, J.L. Ruin estimates under interest force. Insurance:Mathe-matics and Economics,1995,16:7-22
[17]Sundt, B., Teugels, J.L. The adjustment function in ruin estimates under inter-est force. Insurance:Mathe-matics and Economics,1997,19:85-94
[18]Cai, J., Dickson, D.C.M. On the expected discounted penalty function at ruin of a surplus process with interest. Insurance:Mathematics and Economics, 2002,30:389-404
[19]Wu, R., Wang, G, Zhang, C. On a joint distribution for the risk process with constant interest force. Insurance:Mathematics and Economics,2005,36: 365-374
[20]Andersen, E.S. On the collective theory of risk in case of contagion between claims. Bulletin of the Institute of Mathematics and its Applications.1957,12: 257-279
[21]Dickson, D.C.M. On a class of renewal risk process. North American Actuarial Journal,1998,2(3):60-68
[22]Dickson, D.C.M., Hipp, C. Ruin probabilities for Erlang(2) risk process. Insur-ance:Mathematics and Economics,1998,22:251-262
[23]Dickson, D.C.M., Hipp, C. On the of ruin for Erlang(2) risk processes. Insur-ance:Mathematics and Economics,2001,29:333-334
[24]Cheng, Y.B., Tang, Q.H. Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal,2003,7: 1-12
[25]Li, S., Garrido, J. On ruin for Erlang(n) risk process. Insurance:Mathematics and Economics,2004,34:391-408
[26]Willmot, G.E. On the discounted penalty function in the renewal risk model with general interclaim times. Insurance:Mathematics and Economics,2007, 41:17-31
[27]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:Mathe-matics and Economics,1997,21:129-137
[28]Gerber, H.U., Shiu, E.S.W. On the time value of ruin. North American Actu-arial Journal,1998,2:48-72
[29]Lin, X.S., Willmot, G.E. Analysis of a defective renewal equation arising in ruin theory. Insurance:Mathematics and Economics,1999,25:63-84
[30]Willmot,G.E., Dickson,D.C.M. The Gerber-Shiu discounted penalty function in the stationary renewal model. Insurance:Mathematics and Economics,2003, 32:403-411
[31]Pavlova, K.P., Willmot, G.E. The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance:Mathematics and Economics,2004,35:267-277
[32]Gerber, H.U., Shiu, E.S.W. The time value of ruin in a Sparre Andersen model. North American Actuarial Journal,2005,9(2):49-84
[33]Li, S., Lu, Y. On the expected discounted penalty functions for two classes of risk processes. Insurance:Mathematics and Economics,2005,36:179-193
[34]Ahn, S., Badescu, A.L. On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals. Insurance:Mathematics and Economics,2007,41:234-249
[35]Lin, X.S., Willmot, G.E., Drekic, S. The classical risk model with a constant dividend barrier:analysis of the Gerber-Shiu discounted penalty function. Insurance:Mathematics and Economics,2003,33:551-556
[36]Li, S., Garrido, J. On a class of renewal risk models with a constant dividend barrier. Insurance:Mathematics and Economics,2004,35:691-701
[37]Lin, X.S., Pavlova, K.P. The compound Poisson risk model with a threshold dividend strategy. Insurance:Mathematics and Economics,2006,38:57-80
[38]Yuen, K.C., Wang, G.J., Li, W.K. The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier. Insur-ance:Mathematics and Economics,2007,40:104-112
[39]Hu Yang, Zhimin Zhang. Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy. Insurance:Mathematics and Economics,2008,42:984-991
[40]Albrechar, H., Hartinger, J., Tichy, R. On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier. Scandinavian actuarial journal,2005,2:130-136
[41]Gerber H.U. An extension of the renewal equation and its application in the collective theory of risk.Scandinavian Actuarial journal,1970,205-210
[42]Sabine S. Ruin probabilities in perturbed risk models. Insurance:Mathematics and Economics,1998,22:93-104
[43]Chiu, S.N., Yin, C.C. The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance: Mathematics and Economics.2003,33:59-66
[44]Wang G.J., Wu R. A generalization of risk model perturbed by diffusion. Appl.Aath-Juc,1999,14(4):337-341
[45]Wang G. J., Wu R. Some distribution for classical risk process that is perturbed by diffusion.Insurance:Mathematics and Economics,2000,26:15-24
[46]Claudio, M., Giovanni L.T. Asymptotic results for perturbed risk processes with delayed claims. Insurance:Mathematics and Economics,2004,34:307-320
[47]Tsai, C.C.L., Willmot, G.E. On the moments of the surplus process perturbed by diffussion. Insurance:Mathematics and Economics,2002,31:327-350
[48]Gao, H., Yin, C.C. The perturbed Sparre Andersen model with a threshold dividend strategy. Journal of Computational and Applied Mathematics.2008, 220:394-408
[49]邓永录,梁之瞬.随机点过程及其应用.北京：科学出版社,1998
[50]钱敏平,龚光鲁.随机过程论.北京：北京大学出版社,1992
[51]侯振挺,刘万荣,刘再明等.马尔可夫骨架过程-混杂系统模型.湖南：湖南出版社,2000
[52]Asmussen, S. Applied Probability and Queues. John Wiley & Sons,1987
[53]Feller W. An introduction to probability theory and its application. Vol.Ⅱ. New York:John Wiley & Sons,1971
[54]Revuz, D.& Yor, M., Continuous Martingales and Brownian Motion. Sprin-ger, Berlin,1991
[55]何声武,汪嘉冈,严加安.半鞅与随机分析.北京.科学出版社,1995
[56]严加安.鞅与随机分析.上海：上海科学技术出版社,1981
[57]严加安.测度论讲义.北京：科学出版社,2000
[58]Applebaum, D. Lvey processes and stochastic calculus. Cambridge University Press,2004
[59]Ammeter, H. A generalization of the collective theory of risk in regard to fluctuating basic probabilities. Skand. Aktuar Tidskr,1948
[60]Reinhard, J.M. On a class of semi-Markov risk models obtained as classical risk models in a markovian environment. Astin Bulletin ⅩⅣ,1984
[61]Asmussen, S. Risk theory in markovian environment. Scand actuarial J.1989
[62]何树红,徐兴富.双Cox风险模型.云南大学学报,2004,26(4)：275-278
[63]刘艳,胡亦钧.马氏环境下带扰动的Cox相关风险模型破产概率的上界估计.数学年刊,2004,25A(5)：597-586
[64]Li, S., Lu, Y. On the probability of ruin in a Markov-modulated risk model. Insurance:Mathematics and Economics,2005,37:522-532
[65]Neuts, M.F. Algorithms for the waiting time distribution under various queue disciplines in the M/G/1 queue with service time distribution of phase type. Algorithmic Methods in Probability, TIMS Studies in the Management Sciences,1977
[66]Gerber,H.U., Shiu,E.S.W. Discussion of "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process". North American Actuarial Journal,2003(a),7(3):117-119
[67]Gerber,H.U., Shiu,E.S.W. "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process, "Yebin Cheng and Qihe Tang. North American Actuarial Journal,2003(b),7(4):96-101
[68]Shuangming, Li. Discussion of "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process". North American Actuarial Journal,2003,7(3):119-122
[69]Lin, X.S. Discussion of "Moments of the surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process". North American Actuarial Journal, 2003,7(3):122-124
[70]Tsai, C.C.L., Sun, L.J. On the discounted distribution function for the Erlang(2) risk process. Insurance:Mathematics and Economics,2004,35:5-19
[71]Zhao, X.H., Yin, C.C. Ruin problems for a Sparre Andersen risk model.应用概率统计,2005,21(4)：431-442
[72]李俊海.鞅在风险模型中的应用：[博士学位论文].长沙：中南大学,2007
[73]Embrechts, P., Goldie, C.M., Veraverbebe, N. Subexponentiality and infinite divisibility. Probability theory and related fields,1979,49:335-347
[74]Gerber, H.U., Shiu, E.S.W. Discussion of moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal,2003,17(4):96-101
[75]Lin, X.S., Sendova, K.P. The compound Poisson risk model with multiple thre-sholds. Insurance:Mathematics and Economics,2008,42:617-627
[76]Li, S., Garrido, J. The Gerber-Shiu function in a Sparre Andersen risk process perturbed by diffusion. Scandinavian Actuarial Journal,2005,161-186
[77]Xing, Y, Wu, R. The expected valued of a penalty function for a PDMP insur-ance risk model.应用概率统计,2005,21(4)：397-402
[78]Mikhlin, S.G. Integral equations. Pergamon Press, London,1957
[79]Embrechts, P., Kluppelberg, C., Mikosck, T. Modelling Extremal Events. Springer, Berlin,1997
[80]Willmot, G., Lin, X. Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York,2001
[81]Gerber, H.U. Mathematical fun with the compound binomial process. ASTIN Bulletin,1988,18:161-168
[82]Shiu, E.S.W. Probability of eventual ruin in the compound binomial model. ASTIN Bulletin,1988,19:179-190
[83]Willmot,G. E.. A note on a class of delayed renewal risk processes. Insurance: Mathematics and Economics,2004,34:251-257
[84]Zhenhua Bao, Zhongxing Ye. The Gerber-Shiu discounted penalty function in the delayed renewal risk process with random income. Applied Mathematics and Computation.2007,184:857-863
[85]Li, S., Dickson,D.C.M. The maximum surplus before ruin in an Erlang(n) risk process and related problems. Insurance:Mathematics and Economics,2006, 38:529-539
[86]Dufresne, F., Gerber, H.U. Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance:Mathematics and Economics,1991,10: 51-59
[87]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
[88]Tsai, C.C.L., Willmot, G.E. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance:Mathematics and Economics,2002,30:51-66
[89]Tsai, C.C.L. On the expectations of the present values of the time of ruin perturbed by diffusion. Insurance:Mathematics and Economics,2003,32: 413-429
[90]Sarkar, J., Sen, A. Weak convergence approach to compound Poisson risk pro-cesses perturbed by diffusion. Insurance:Mathematics and Economics,2005, 36:421-432