常利率下风险模型破产问题的研究
|
摘要
经典的风险模型是考虑理赔次数过程为泊松过程，个别理赔额序列独立同分布且与理赔次数过程相互独立，保费率为常数的情形。在此模型下，当个别理赔额服从指数分布的时候，Filip Lundberg和Cramer等人得到了破产概率的显示表达式。此外，利用William Feller介绍的更新理论的方法，他们得到了破产概率的指数上界，Gerber(1973)利用鞅的方法也得到了同样的结果。关于破产严重性问题近来引起了广泛的关注。Dufresne和Gerber(1988a,b)、Gerber和Shiu(1997,1998)、Gerber等(1987)、Willmot和Lin(1997)以及Yang和Zhang(2001a,b)等都就破产时刻、破产前瞬时盈余和破产时赤字的分布进行了分析。
经典风险模型没有考虑到利率因素的影响。在实际操作中，保险公司的大部分盈余来自于投资的收入，所以有固定利率的风险模型正日益受到人们的关注。Sundt和Teugels(1995)研究了常利率下复合泊松模型的终极破产概率，而且在个别理赔额服从指数分布的特殊情形下，他们还得到了终极破产概率的显式解。Yang(1999)考虑了常利率下离散时间风险模型，利用鞅的方法得到了Lundberg型不等式以及破产概率的非指数型上界。在常利率且有随机投资收入的假设下，Paulsen和Gjessing(1997)得到了Lundberg型不等式。
本文主要考虑常利率下的风险模型，对破产严重性、破产概率的上界以及再保险中的自留额等问题进行了分析。具体来说包括以下几方面的内容：
当初始准备金为u时，第一部分引进了与常数利率δ和Laplace变换自变量α相关的罚金折现期望值Φ_(δ，α)(u)。利用更新理论的方法得到Φ_(δ，α)(u)满足的一个积分方程，又利用Laplace变换的技巧得到了该期望值的初始值Φ_(δ，α)(0)的精确解，从而给出罚金折现期望值应满足的解的形式。在此基础上，用分析的方法讨论了破产前
华东师范大学博士学位论文口004)
瞬时盈余、破产时的赤字和破产时刻联合与边际分布的折现期望值，并得到它们之
I司满足的一个关系式，它推广了Di改son(1992)、eerber和slliu(1995)以及cai和
Di(:k soll(20o2)中的结果.此外还分析了破产前瞬时盈余、破产时的赤字和破产时刻
的联合与边际矩的性质.
假设保单到达时间间隔服从指数分布，理赔次数过程为一般更新过程，且保单
到达过程与理赔过程相互独立，称之为泊松一更新风险模型.第二部分通过构造离
散上鞍和利用递归的方法，分别得到泊松一更新风险模型中终极破产概率的两种上
界.在具体实例中，通过模拟计算对这两种方法进行了比较，并说明了它们的优劣
性.
考虑有息力的SParre Andersoll风险模型在有限时间内破产概率的上界问题.
Sl〕arreA:Iclerson(1957)研究了理赔为一般更新到达风险模型的终极破产概率，此
后，理赔为非Poisson到达风险模型的研究得到了极大的关注.Malillovskii(1995)和
wallg(2002)分别研究了sparre Anderson模型中有限时间内破产概率的Lal)la(:e变
换，当个别理赔额服从指数分布或混合指数分布时，他们给出了相应的Lal)la(二变
换的显示表达式.第三部分通过构造一个连续上鞍得到了有息力的sParreAl1del’so:1
风险模型在有限时间内破产概率的上界.当理赔时间间隔服从指数分布、混合指数
分布以及Erlang(2)分布等常见分布时，相应破产概率的上界作为特例情形得到.
最后，在有息力的更新风险模型的基础上，我们考虑了再保险的影响.这里假
设再保费的计算采用期望值原理，其类别属于超额赔款再保险.通过研究有息力的
Sl)a1T。Al，。lerson风险模型在有限时间内破产概率的上界与自留额的关系，利用第三
部分的结论，在使其破产概率的上界达到最小的意义下确定保方的自留额.
In the classical risk model, the number of claims from an insurance portfolio is assumed to follow a Poission process , the individual claim sizes are independent and identical random variables, and the premiums are described by a constant rate of income. In this kind of model, the clear expression for the ruin probability is given by Filip Lundberg and Cramer when the claim amount is exponentially distributed. Futhermore, they get the exponential upper bound by the aid of a renewal technique introduced by William Feller, which is also proved by Gerber(1973) using a martingale approach. The problem on the severity of ruin has recently received a remarkable attention. In Dufresne and Gerber(1988a,b),Gerber and Shiu(1997,1998),Gerber et al.(1987), Willmot and Lin(1997), and Yang and Zhang(2001a,b), the distributions of the ruin time, the surplus immediately prior to ruin and the deficit at ruin were considered.
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. In recent years , the risk models with deterministic interest rate have been paid more attention. Sundt and Teugels(1995) considered the ultimate ruin probability in a compound Poisson model with a constant interest force, and they get its exact solution at the special case of exponential claim sizes. Yang(1999) considered a discrete time risk model with a constant interest force and both Lundberg-type inequality and non-exponential upper bounds for ruin probabilities were obtained by using martingale inequalities. Under the assumption of stochastic investment income , but a constant interest rate, a Lundberg-type inequality was obtained in Paulsen and Gjessing(1997).
This thesis is devoted to a study of severity of ruin , upper bounds for ruin probability
and retention levels for reinsurance in the risk models with a constant interest force. Concretely, four aspects of Works are considered:
In the first part, we consider the expected value of a discounted function(u) associated with a constant force and the argument a in a Laplace transforms as a function of the initial surplus u .By using the techniques of renewal theory , we derive an integral equation for (u).We then find an exact solution for (0) .Therefore, the form of the solution for the expected value of a discounted function can be found.Hereafter, we consider the expected value of a discounted function connected with the joint and marginal distributions of the surplus immediately prior to ruin , the deficit at ruin and the ruin time, and obtain a relation among them, which generalise the results of Dickson(1992),Gerber and Shiu(1998) and Cai and Dickson(2002).Furthermore, we analysis of the properties of the joint and marginal moments of the surplus immediately prior to ruin , the deficit at ruin and the ruin time.
Poisson-renewal risk model is such that the inter-occurrence time of guarantee slips' arrivals are independ and indentical exponential random variables , the number of claims follows an ordinary renewal process, and the number of guarantee slips is independ of that of claims . In the second part, two kinds of upper bounds in the model are obtained by martingale and recursive techniques respectively, and the numerical comparisons of upper bounds derived by each technique are presented.
The upper bounds for the finite time ruin probability in Sparre Anderson risk model are derived.Sparre Anderson(1957) considered the ultimate ruin probability when claims occur as a general renewal process.Since then, non-Poissonian risk models have been drawn more attention.Malinovskii(1998) and Wang(2001) considered Laplace transform of the finite time ruin probability in Sparre Anderson risk model, and they give clear expression
for the Laplace transforms when the individual claim sizes are exponentially and mixed exponentially distributed. In the third part, the upper bounds for the finite time ruin probability in Sparre Anderson risk model with a con
经典风险模型没有考虑到利率因素的影响。在实际操作中，保险公司的大部分盈余来自于投资的收入，所以有固定利率的风险模型正日益受到人们的关注。Sundt和Teugels(1995)研究了常利率下复合泊松模型的终极破产概率，而且在个别理赔额服从指数分布的特殊情形下，他们还得到了终极破产概率的显式解。Yang(1999)考虑了常利率下离散时间风险模型，利用鞅的方法得到了Lundberg型不等式以及破产概率的非指数型上界。在常利率且有随机投资收入的假设下，Paulsen和Gjessing(1997)得到了Lundberg型不等式。
本文主要考虑常利率下的风险模型，对破产严重性、破产概率的上界以及再保险中的自留额等问题进行了分析。具体来说包括以下几方面的内容：
当初始准备金为u时，第一部分引进了与常数利率δ和Laplace变换自变量α相关的罚金折现期望值Φ_(δ，α)(u)。利用更新理论的方法得到Φ_(δ，α)(u)满足的一个积分方程，又利用Laplace变换的技巧得到了该期望值的初始值Φ_(δ，α)(0)的精确解，从而给出罚金折现期望值应满足的解的形式。在此基础上，用分析的方法讨论了破产前
华东师范大学博士学位论文口004)
瞬时盈余、破产时的赤字和破产时刻联合与边际分布的折现期望值，并得到它们之
I司满足的一个关系式，它推广了Di改son(1992)、eerber和slliu(1995)以及cai和
Di(:k soll(20o2)中的结果.此外还分析了破产前瞬时盈余、破产时的赤字和破产时刻
的联合与边际矩的性质.
假设保单到达时间间隔服从指数分布，理赔次数过程为一般更新过程，且保单
到达过程与理赔过程相互独立，称之为泊松一更新风险模型.第二部分通过构造离
散上鞍和利用递归的方法，分别得到泊松一更新风险模型中终极破产概率的两种上
界.在具体实例中，通过模拟计算对这两种方法进行了比较，并说明了它们的优劣
性.
考虑有息力的SParre Andersoll风险模型在有限时间内破产概率的上界问题.
Sl〕arreA:Iclerson(1957)研究了理赔为一般更新到达风险模型的终极破产概率，此
后，理赔为非Poisson到达风险模型的研究得到了极大的关注.Malillovskii(1995)和
wallg(2002)分别研究了sparre Anderson模型中有限时间内破产概率的Lal)la(:e变
换，当个别理赔额服从指数分布或混合指数分布时，他们给出了相应的Lal)la(二变
换的显示表达式.第三部分通过构造一个连续上鞍得到了有息力的sParreAl1del’so:1
风险模型在有限时间内破产概率的上界.当理赔时间间隔服从指数分布、混合指数
分布以及Erlang(2)分布等常见分布时，相应破产概率的上界作为特例情形得到.
最后，在有息力的更新风险模型的基础上，我们考虑了再保险的影响.这里假
设再保费的计算采用期望值原理，其类别属于超额赔款再保险.通过研究有息力的
Sl)a1T。Al，。lerson风险模型在有限时间内破产概率的上界与自留额的关系，利用第三
部分的结论，在使其破产概率的上界达到最小的意义下确定保方的自留额.
In the classical risk model, the number of claims from an insurance portfolio is assumed to follow a Poission process , the individual claim sizes are independent and identical random variables, and the premiums are described by a constant rate of income. In this kind of model, the clear expression for the ruin probability is given by Filip Lundberg and Cramer when the claim amount is exponentially distributed. Futhermore, they get the exponential upper bound by the aid of a renewal technique introduced by William Feller, which is also proved by Gerber(1973) using a martingale approach. The problem on the severity of ruin has recently received a remarkable attention. In Dufresne and Gerber(1988a,b),Gerber and Shiu(1997,1998),Gerber et al.(1987), Willmot and Lin(1997), and Yang and Zhang(2001a,b), the distributions of the ruin time, the surplus immediately prior to ruin and the deficit at ruin were considered.
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. In recent years , the risk models with deterministic interest rate have been paid more attention. Sundt and Teugels(1995) considered the ultimate ruin probability in a compound Poisson model with a constant interest force, and they get its exact solution at the special case of exponential claim sizes. Yang(1999) considered a discrete time risk model with a constant interest force and both Lundberg-type inequality and non-exponential upper bounds for ruin probabilities were obtained by using martingale inequalities. Under the assumption of stochastic investment income , but a constant interest rate, a Lundberg-type inequality was obtained in Paulsen and Gjessing(1997).
This thesis is devoted to a study of severity of ruin , upper bounds for ruin probability
and retention levels for reinsurance in the risk models with a constant interest force. Concretely, four aspects of Works are considered:
In the first part, we consider the expected value of a discounted function(u) associated with a constant force and the argument a in a Laplace transforms as a function of the initial surplus u .By using the techniques of renewal theory , we derive an integral equation for (u).We then find an exact solution for (0) .Therefore, the form of the solution for the expected value of a discounted function can be found.Hereafter, we consider the expected value of a discounted function connected with the joint and marginal distributions of the surplus immediately prior to ruin , the deficit at ruin and the ruin time, and obtain a relation among them, which generalise the results of Dickson(1992),Gerber and Shiu(1998) and Cai and Dickson(2002).Furthermore, we analysis of the properties of the joint and marginal moments of the surplus immediately prior to ruin , the deficit at ruin and the ruin time.
Poisson-renewal risk model is such that the inter-occurrence time of guarantee slips' arrivals are independ and indentical exponential random variables , the number of claims follows an ordinary renewal process, and the number of guarantee slips is independ of that of claims . In the second part, two kinds of upper bounds in the model are obtained by martingale and recursive techniques respectively, and the numerical comparisons of upper bounds derived by each technique are presented.
The upper bounds for the finite time ruin probability in Sparre Anderson risk model are derived.Sparre Anderson(1957) considered the ultimate ruin probability when claims occur as a general renewal process.Since then, non-Poissonian risk models have been drawn more attention.Malinovskii(1998) and Wang(2001) considered Laplace transform of the finite time ruin probability in Sparre Anderson risk model, and they give clear expression
for the Laplace transforms when the individual claim sizes are exponentially and mixed exponentially distributed. In the third part, the upper bounds for the finite time ruin probability in Sparre Anderson risk model with a con
引文
[1] 程宗毛(1999)破产时刻罚金折现期望值．应用概率统计，15(3)：225-233．
[2] 孙立娟，顾岚(2000)保险公司破产概率的估计及随机模拟．系统工程理论与实践，20(7)：63-68，
[3] 吴荣，杜勇宏(2002)常利率下的更新风险模型．工程数学学报，19(1)：46-54．
[4] 谢志刚，韩天雄(2000)风险理论与非寿险精算．南开大学出版社．
[5] Andersen,E. Sparre(1957) On the collective theory of risk in the case of contagion between the claims. Transactions ⅩⅤth International Congress of Actuaries, 2 : 219-229.
[6] Asmussen S.(2000) Ruin probabilities. World Scientific, Singapore.
[7] Asmussen S.(2002) Erlangian approximation for finite-horizon ruin probabilities. ASTIN Bulletin, 32: 267-281.
[8] Von Bahr B.(1974) Ruinprobabilities expresed in terms of ladder height distributions. Scandinavian Actuarial Journal: 190-204.
[9] Boogart P.,Crijns V.(1987) Upper bound on ruin probabilities in case of negative loadings and positive interest rate. Insurance: Mathematics and Economics, 6: 221-232.
[10] Boogaert P., De Waegenaere A.(1990) Macro-economic version of a classical formula in risk theory. Insurance: Mathematics and Economics, 9: 155-162.
[11] Boogaert P., Haezendonck J.(1989) Delay in claim settlement. Insurance : Mathematics and Economics, 8: 321-330.
[12] Brekelmans,R.,Waegenaere,A.D.(2001) Approximating the finite-time ruin probability under interest force. Insurance: Mathematics and Economics, 29 : 217-229.
[13] Cai J.,Dicson D.C.(2002) On the expected discounted penalty function at ruin of a surplus process with interest. Insurance : Mathematics and Economics, 30 : 389-404.
[14] Cai J.,Dicson D.C.(2003) Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest. Insurance : Mathematics and Economics, 32 : 61-71.
[15] Centeno M.L.(1985) On combining quota-share and excess of loss. ASTIN Bulletin , 15:
49-63.
[16] Centeno M.L.(1986) Measuring the effects of reinsurance by the adjustment coefficient. Insurance: Mathematics and Economics, 5: 169-182.
[17] Centeno M.L.(1997) Excess of loss reinsurance and the probability of ruin in finite horizon. ASTIN Bulletin, 27: 59-70.
[18] Centeno M.L.(2002a) Measuring the effects of reinsurance by the adjustment cofficient in the Sparre Anderson model. Insurance: Mathematics and Economics, 30 : 37-49.
[19] Centeno M.L.(2002b) Excess of loss reinsurance and Gerber's inequality in the Sparre Anderson model. Insurance: Mathematics and Economics, 31: 415-427.
[20] Cheng S.,Gerber H.U.,Shiu E.S.(2000) Discounted probabilites and ruin theory in the compound binomial model. Insurance : Mathematics and Economics, 26: 239-250.
[21] Cromer H.(1945) Mathematical Methods of Statistics. Almqvist & Wiksell, Stockholm and Princeton University Press, Princeton.
[22] Dassios A., Embrechts P.(1989) Martingales and insurance risk. Commun. Statist. -Stochastic models, 5: 181-217.
[23] Davis M.H.(1984) Piecewise-determinisic Markov processes: A general class of non-diffusion stochastic models. J.R.Statist.Soc.B. , 46: 353-388.
[24] Davis M.H.(1993) Markov Models and Optimization. Chapman & Hall, London.
[25] De Vylder F., Goovaerts M.J.(1988) Recursive calculation of finite-time ruin probabilities. Insurance: Mathematics and Economics, 7: 1-7.
[26] De Vylder F., Goovaerts M.J.(1998) Discussison of 'The time value of ruin' by Gerber and Shiu. North American Actuarial Journal, 2: 72-74.
[27] Delbaen F.,Haezendonck J.(1987) Classical risk theory in an economic environment. Insurance: Mathematics and Economics, 6 : 85-116.
[28] Delbaen F.(1990) A remark on the moments of ruin time in classical risk theory. Insurance: Mathematics and Economics, 9: 121-126.
[29] Dickson D.C.(1991) Recursive calculation of survival probability. ASTIN Bulletin, 21: 199-221.
[30] Dickson D.C.(1992) On the distribution of the surplus prior to ruin. Insurance: Mathematics and Economics, 11: 191-207.
[31] Dickson D.C.,Dos Reis A.D.(1996) On the distribution of the duration of negative surplus. Scandinavian Actuarial Journal: 148-164.
[32] Dickson D.C.,Dos Reis A.D.,Water H.R.(1995) Some stable algorithms in ruin theory and application. ASTIN Bulletin, 25: 153-175.
[33] Dickson D.C.(1998) On a class of renewal risk processes. North American Actuarial Journal, 2 : 60-73.
[34] Dickson D,C., Water H.R.(1992) The probability and severity of ruin in finite and in infinite time. ASTIN Bulletin, 22: 177-190.
[35] Dickson D.C., Waters H.R.(1999) Ruin probabilities with compounding assets. Insurance: Mathmatics and Economics, 25: 49-62.
[36] Dufresne F.,Gerber H.U.(1988a) The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics, 7: 193-199.
[37] Dufresne F.,Gerber H.U.(1988b) The probability and severity of ruin for combinations of exponential claim amount distributions and their translations. Insurance : Mathematics and Economics, 7: 75-80.
[38] Embrechts P.,Veraverbeke N.(1982) Estimates for the probability of ruin with speciaiemphasis on the possilbility of large claims. Insurance: Mathematics and Economics, 1: 55-72.
[39] Fagiuoli E.,Pellerey F.(1994) Preservation of certain classes of life destributions under Poisson shock models. Journal of Applied Probability, 31: 458-465.
[40] Feller W.(1971) An Introduction to Probability Theory and its applications, 2nd Edition. John Wiley & Sons, New York.
[41] Gerber H.U.(1973) Martingales in risk theory. Mitt. Vet. Schweiz. Vers. Math., 73: 205-216.
[42] Gerber H.U.(1976) An Introduction to Mathematical Risk Theory. S.S.Heubner Foundation Monograph Series 8, Philadelphia.
[43] Gerber H.U.,Goovaerts M.J.,Kaas R.(1987) On the probability and severity of ruin. ASTIN Bulletin, 17: 151-163.
[44] Gerber H.U.,Shiu E.S. (1997) The joint distribution of the time of ruin,the surplus immediately before ruin,and the deficit at ruin. Insurance : Mathematics and Economics, 21: 129-137.
[45] Gerber H.U.,Shiu E.S.(1998) On the time value of ruin. North American Actuarial Journal, 2 : 48-78.
[46] Gerber H., Shiu E.S.(1999) From ruin theory to pricing reset guarantees and perpetual put options. Insurance: Mathematics and Economics, 24: 3-14.
[47] Gjessing H.K.,Paulsen J.(1997) Prensent value distributions with applications to ruin theory and stochastic equations. Stochastic Processes and Their Applications, 71: 123-144.
[48] Grandell J.(1977) A class of approximations of ruin probabilities. Scandinavian Actuarial Journal: 38-52.
[49] Grandell J.(1991) Aspects of Risk Theory. Springer,New York.
[50] Herkenrath U.(1986) On the extimation of th eadjustment coefficient in tisk theory by means of stochastic approximation procedures. Insurance: Mathematics and Economics, 5: 305-313.
[51] Hesselager 0.(1990) Some results on optimal reinsurance in terms of the adjustment coneffi-cient. Scandinavian Actuarial Journal: 80-95.
[52] Hesselager O.,Wang D.,Willmot G.E.(1998) Exponential and scale mixtures and equilibrium distributium distributions. Scandinavian Actuarial Journal: 125-142.
[53] Lamberton D., Lapeyre B.(1996) Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall,London.
[54] Lin X.(1996) Tail of compound distributions and excess time. Journal of Applied Probability, 33: 184-195.
[55] Lin X.,Willmot G.(1999) Analysis of a defective renewal equation arising in ruin theory. Insurance: Mathematics and Economics, 25 : 63-84.
[56] Lin X.,Willmot G.E.(2000) The moment of the time of ruin ,the surplus before ruin,and the deficit at ruin. Insurance : Mathematics and Economics, 27: 19-44.
[57] Malinovskii V.K.(1994) Corrected normal approximtion for the probability of ruin within finite time. Scandinavian Actuarial Journal: 161-174.
[58] Malinovskii V.K.(1998) Non-Poissonian claims' arrivals and calcultion of the probability of ruin. Insurance : Mathematics and Economics, 22: 123-138.
[59] Massey W.,Whitt W.(1993) A probabilistic generalisation of Taylor's theorem. Statistics and Probability Letters, 16: 51-54.
[60] Miklin S.(1957) Integral Equations. Pergamon Press, London.
[61] Panjer H.(1981) Recursive calculation of a family of compound distributions. ASTIN Bulletin, 12: 22-26.
[62] Paulsen J.(1998) Ruin theory with compounding assets: a survey. Insurance: Mathematics and Economics, 22: 3-16.
[63] Paulsen J.,Gjessing H.K.(1997) Ruin theory with stochastic return on investments. Advances in Applied Probability, 29: 965-985.
[64] Picard Ph.,Lefevre C.(1998) The moments of ruin time in the classical risk model with discrete claim size distribution. Insurance: Mathematics and Economics, 23: 157-172.
[65] Picard Ph.,Lefevre C.(1999) Corrigendum to the moments of ruin time in the classical risk model with discrete claim size distribution. Insurance: Mathematics and Economics, 25: 105-107.
[66] Reinhard J.M.(1984) On a class of semi-Markov risk models obtaimed as classical risk models in a markovian enviroament. ASTIN Bulletin, 9: 23-43.
[67] Rolski T., Schmidli H. , Schmidt V., Teugels J.(1999) Stochastic Processes for Insurance and Finance. John Wiley & Sons, New York.
[68] Rosenlund S.(1989) Numerical calculation of the Cramér-Lundberg approximation. Scandinavian Actuarial Journal: 119-122.
[69] Ross R.(1996) Stochastic Processes , end Edition. John Wiley & Sons, New York.
[70] Schmidli H.(1999) On the distribution of the surplus prior to and at ruin. ASTIN Bulletin, 29 : 227-244.
[71] Shiu E.S.(1988) Calculation of the probability of eventual ruin by Beekman's convolution series. Insurance: Mathematics and Economics, 7: 41-47.
[72] Sundt B.,Teugels J.L.(1995) Ruin estimates under interest force. Insurance: Mathematics
and Economics, 16: 7-22.
[73] Sundt B.,Teugels J.L.(1997) The adjustment function in ruin estimates under interest force. Insurance: Mathematics and Economics, 19: 85-94.
[74] Thorin O.(1971) Analytical steps towards a numerical calculation of the ruin probability for a finite period when the risk process is of the Poisson type of the more general type studied by Sparre Andersen. ASTIN Bulletin, 6: 137-153.
[75] Thorin O.(1975) Stationarity aspects of the Sparre Andersen risk process and the corresponding ruin probabilities. Scandinavian Actuarial Journal: 87-98.
[76] Thorin O.(1982) Probabilities of ruin. Scandinavian Actuarial Journal: 65-102.
[77] Vàzquez-Abad F.(2000) RPA pathwise derivative estimation of ruin probabilities. Insurance: Mathematics and Economics, 26: 269-288.
[78] Wang R.,Liu H.(2002) On the ruin probability under a class of risk processes. ASTIN Bulletin, 32: 81-90.
[79] Waters H.(1979) Excess of loss reinsurance limits. Scandinavian Actuarial Journal: 37-43.
[80] Waters H.(1983) Some mathematical aspects of reinsurance. Insurance : Mathematics and Economics, 2: 17-26.
[81] Wikstad N..(1983) A numerical illustration of differences between ruin probabilities originated in the ordinary and in the stationary cases. Scandinavian Actuarial Journal: 47-48.
[82] Willmot G,E.(1999) A Laplace transform representation in a class of renewal queueing and risk processes. Journal of Applied Probability, 36: 570-584.
[83] Willmot G.E.,Lin X.S.(1997) Exact and approximate properties of the distribution of surplus before and afer ruin. Insurance: Mathematics and Economics, 23: 91-110.
[84] Willmot G.E.,Lin X.S.(2000) Lundberg Approximations for Compound Distribution with Insurance Applications. Springer,New York.
[85] Yang H.(1999) Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal: 66-79.
[86] Yang H., Zhang L.(2001a) The joint distribution of surplus immediately before ruin and the deficit at ruin under interest force. North American Actuarial Journal, 5 : 92-103.
[87] Yang H. Zhang L.(2001b) On the distribution of the surplus immediately after ruin under interest force. Insurance: Mathematics and Economics, 29: 247-256.
[2] 孙立娟，顾岚(2000)保险公司破产概率的估计及随机模拟．系统工程理论与实践，20(7)：63-68，
[3] 吴荣，杜勇宏(2002)常利率下的更新风险模型．工程数学学报，19(1)：46-54．
[4] 谢志刚，韩天雄(2000)风险理论与非寿险精算．南开大学出版社．
[5] Andersen,E. Sparre(1957) On the collective theory of risk in the case of contagion between the claims. Transactions ⅩⅤth International Congress of Actuaries, 2 : 219-229.
[6] Asmussen S.(2000) Ruin probabilities. World Scientific, Singapore.
[7] Asmussen S.(2002) Erlangian approximation for finite-horizon ruin probabilities. ASTIN Bulletin, 32: 267-281.
[8] Von Bahr B.(1974) Ruinprobabilities expresed in terms of ladder height distributions. Scandinavian Actuarial Journal: 190-204.
[9] Boogart P.,Crijns V.(1987) Upper bound on ruin probabilities in case of negative loadings and positive interest rate. Insurance: Mathematics and Economics, 6: 221-232.
[10] Boogaert P., De Waegenaere A.(1990) Macro-economic version of a classical formula in risk theory. Insurance: Mathematics and Economics, 9: 155-162.
[11] Boogaert P., Haezendonck J.(1989) Delay in claim settlement. Insurance : Mathematics and Economics, 8: 321-330.
[12] Brekelmans,R.,Waegenaere,A.D.(2001) Approximating the finite-time ruin probability under interest force. Insurance: Mathematics and Economics, 29 : 217-229.
[13] Cai J.,Dicson D.C.(2002) On the expected discounted penalty function at ruin of a surplus process with interest. Insurance : Mathematics and Economics, 30 : 389-404.
[14] Cai J.,Dicson D.C.(2003) Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest. Insurance : Mathematics and Economics, 32 : 61-71.
[15] Centeno M.L.(1985) On combining quota-share and excess of loss. ASTIN Bulletin , 15:
49-63.
[16] Centeno M.L.(1986) Measuring the effects of reinsurance by the adjustment coefficient. Insurance: Mathematics and Economics, 5: 169-182.
[17] Centeno M.L.(1997) Excess of loss reinsurance and the probability of ruin in finite horizon. ASTIN Bulletin, 27: 59-70.
[18] Centeno M.L.(2002a) Measuring the effects of reinsurance by the adjustment cofficient in the Sparre Anderson model. Insurance: Mathematics and Economics, 30 : 37-49.
[19] Centeno M.L.(2002b) Excess of loss reinsurance and Gerber's inequality in the Sparre Anderson model. Insurance: Mathematics and Economics, 31: 415-427.
[20] Cheng S.,Gerber H.U.,Shiu E.S.(2000) Discounted probabilites and ruin theory in the compound binomial model. Insurance : Mathematics and Economics, 26: 239-250.
[21] Cromer H.(1945) Mathematical Methods of Statistics. Almqvist & Wiksell, Stockholm and Princeton University Press, Princeton.
[22] Dassios A., Embrechts P.(1989) Martingales and insurance risk. Commun. Statist. -Stochastic models, 5: 181-217.
[23] Davis M.H.(1984) Piecewise-determinisic Markov processes: A general class of non-diffusion stochastic models. J.R.Statist.Soc.B. , 46: 353-388.
[24] Davis M.H.(1993) Markov Models and Optimization. Chapman & Hall, London.
[25] De Vylder F., Goovaerts M.J.(1988) Recursive calculation of finite-time ruin probabilities. Insurance: Mathematics and Economics, 7: 1-7.
[26] De Vylder F., Goovaerts M.J.(1998) Discussison of 'The time value of ruin' by Gerber and Shiu. North American Actuarial Journal, 2: 72-74.
[27] Delbaen F.,Haezendonck J.(1987) Classical risk theory in an economic environment. Insurance: Mathematics and Economics, 6 : 85-116.
[28] Delbaen F.(1990) A remark on the moments of ruin time in classical risk theory. Insurance: Mathematics and Economics, 9: 121-126.
[29] Dickson D.C.(1991) Recursive calculation of survival probability. ASTIN Bulletin, 21: 199-221.
[30] Dickson D.C.(1992) On the distribution of the surplus prior to ruin. Insurance: Mathematics and Economics, 11: 191-207.
[31] Dickson D.C.,Dos Reis A.D.(1996) On the distribution of the duration of negative surplus. Scandinavian Actuarial Journal: 148-164.
[32] Dickson D.C.,Dos Reis A.D.,Water H.R.(1995) Some stable algorithms in ruin theory and application. ASTIN Bulletin, 25: 153-175.
[33] Dickson D.C.(1998) On a class of renewal risk processes. North American Actuarial Journal, 2 : 60-73.
[34] Dickson D,C., Water H.R.(1992) The probability and severity of ruin in finite and in infinite time. ASTIN Bulletin, 22: 177-190.
[35] Dickson D.C., Waters H.R.(1999) Ruin probabilities with compounding assets. Insurance: Mathmatics and Economics, 25: 49-62.
[36] Dufresne F.,Gerber H.U.(1988a) The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics, 7: 193-199.
[37] Dufresne F.,Gerber H.U.(1988b) The probability and severity of ruin for combinations of exponential claim amount distributions and their translations. Insurance : Mathematics and Economics, 7: 75-80.
[38] Embrechts P.,Veraverbeke N.(1982) Estimates for the probability of ruin with speciaiemphasis on the possilbility of large claims. Insurance: Mathematics and Economics, 1: 55-72.
[39] Fagiuoli E.,Pellerey F.(1994) Preservation of certain classes of life destributions under Poisson shock models. Journal of Applied Probability, 31: 458-465.
[40] Feller W.(1971) An Introduction to Probability Theory and its applications, 2nd Edition. John Wiley & Sons, New York.
[41] Gerber H.U.(1973) Martingales in risk theory. Mitt. Vet. Schweiz. Vers. Math., 73: 205-216.
[42] Gerber H.U.(1976) An Introduction to Mathematical Risk Theory. S.S.Heubner Foundation Monograph Series 8, Philadelphia.
[43] Gerber H.U.,Goovaerts M.J.,Kaas R.(1987) On the probability and severity of ruin. ASTIN Bulletin, 17: 151-163.
[44] Gerber H.U.,Shiu E.S. (1997) The joint distribution of the time of ruin,the surplus immediately before ruin,and the deficit at ruin. Insurance : Mathematics and Economics, 21: 129-137.
[45] Gerber H.U.,Shiu E.S.(1998) On the time value of ruin. North American Actuarial Journal, 2 : 48-78.
[46] Gerber H., Shiu E.S.(1999) From ruin theory to pricing reset guarantees and perpetual put options. Insurance: Mathematics and Economics, 24: 3-14.
[47] Gjessing H.K.,Paulsen J.(1997) Prensent value distributions with applications to ruin theory and stochastic equations. Stochastic Processes and Their Applications, 71: 123-144.
[48] Grandell J.(1977) A class of approximations of ruin probabilities. Scandinavian Actuarial Journal: 38-52.
[49] Grandell J.(1991) Aspects of Risk Theory. Springer,New York.
[50] Herkenrath U.(1986) On the extimation of th eadjustment coefficient in tisk theory by means of stochastic approximation procedures. Insurance: Mathematics and Economics, 5: 305-313.
[51] Hesselager 0.(1990) Some results on optimal reinsurance in terms of the adjustment coneffi-cient. Scandinavian Actuarial Journal: 80-95.
[52] Hesselager O.,Wang D.,Willmot G.E.(1998) Exponential and scale mixtures and equilibrium distributium distributions. Scandinavian Actuarial Journal: 125-142.
[53] Lamberton D., Lapeyre B.(1996) Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall,London.
[54] Lin X.(1996) Tail of compound distributions and excess time. Journal of Applied Probability, 33: 184-195.
[55] Lin X.,Willmot G.(1999) Analysis of a defective renewal equation arising in ruin theory. Insurance: Mathematics and Economics, 25 : 63-84.
[56] Lin X.,Willmot G.E.(2000) The moment of the time of ruin ,the surplus before ruin,and the deficit at ruin. Insurance : Mathematics and Economics, 27: 19-44.
[57] Malinovskii V.K.(1994) Corrected normal approximtion for the probability of ruin within finite time. Scandinavian Actuarial Journal: 161-174.
[58] Malinovskii V.K.(1998) Non-Poissonian claims' arrivals and calcultion of the probability of ruin. Insurance : Mathematics and Economics, 22: 123-138.
[59] Massey W.,Whitt W.(1993) A probabilistic generalisation of Taylor's theorem. Statistics and Probability Letters, 16: 51-54.
[60] Miklin S.(1957) Integral Equations. Pergamon Press, London.
[61] Panjer H.(1981) Recursive calculation of a family of compound distributions. ASTIN Bulletin, 12: 22-26.
[62] Paulsen J.(1998) Ruin theory with compounding assets: a survey. Insurance: Mathematics and Economics, 22: 3-16.
[63] Paulsen J.,Gjessing H.K.(1997) Ruin theory with stochastic return on investments. Advances in Applied Probability, 29: 965-985.
[64] Picard Ph.,Lefevre C.(1998) The moments of ruin time in the classical risk model with discrete claim size distribution. Insurance: Mathematics and Economics, 23: 157-172.
[65] Picard Ph.,Lefevre C.(1999) Corrigendum to the moments of ruin time in the classical risk model with discrete claim size distribution. Insurance: Mathematics and Economics, 25: 105-107.
[66] Reinhard J.M.(1984) On a class of semi-Markov risk models obtaimed as classical risk models in a markovian enviroament. ASTIN Bulletin, 9: 23-43.
[67] Rolski T., Schmidli H. , Schmidt V., Teugels J.(1999) Stochastic Processes for Insurance and Finance. John Wiley & Sons, New York.
[68] Rosenlund S.(1989) Numerical calculation of the Cramér-Lundberg approximation. Scandinavian Actuarial Journal: 119-122.
[69] Ross R.(1996) Stochastic Processes , end Edition. John Wiley & Sons, New York.
[70] Schmidli H.(1999) On the distribution of the surplus prior to and at ruin. ASTIN Bulletin, 29 : 227-244.
[71] Shiu E.S.(1988) Calculation of the probability of eventual ruin by Beekman's convolution series. Insurance: Mathematics and Economics, 7: 41-47.
[72] Sundt B.,Teugels J.L.(1995) Ruin estimates under interest force. Insurance: Mathematics
and Economics, 16: 7-22.
[73] Sundt B.,Teugels J.L.(1997) The adjustment function in ruin estimates under interest force. Insurance: Mathematics and Economics, 19: 85-94.
[74] Thorin O.(1971) Analytical steps towards a numerical calculation of the ruin probability for a finite period when the risk process is of the Poisson type of the more general type studied by Sparre Andersen. ASTIN Bulletin, 6: 137-153.
[75] Thorin O.(1975) Stationarity aspects of the Sparre Andersen risk process and the corresponding ruin probabilities. Scandinavian Actuarial Journal: 87-98.
[76] Thorin O.(1982) Probabilities of ruin. Scandinavian Actuarial Journal: 65-102.
[77] Vàzquez-Abad F.(2000) RPA pathwise derivative estimation of ruin probabilities. Insurance: Mathematics and Economics, 26: 269-288.
[78] Wang R.,Liu H.(2002) On the ruin probability under a class of risk processes. ASTIN Bulletin, 32: 81-90.
[79] Waters H.(1979) Excess of loss reinsurance limits. Scandinavian Actuarial Journal: 37-43.
[80] Waters H.(1983) Some mathematical aspects of reinsurance. Insurance : Mathematics and Economics, 2: 17-26.
[81] Wikstad N..(1983) A numerical illustration of differences between ruin probabilities originated in the ordinary and in the stationary cases. Scandinavian Actuarial Journal: 47-48.
[82] Willmot G,E.(1999) A Laplace transform representation in a class of renewal queueing and risk processes. Journal of Applied Probability, 36: 570-584.
[83] Willmot G.E.,Lin X.S.(1997) Exact and approximate properties of the distribution of surplus before and afer ruin. Insurance: Mathematics and Economics, 23: 91-110.
[84] Willmot G.E.,Lin X.S.(2000) Lundberg Approximations for Compound Distribution with Insurance Applications. Springer,New York.
[85] Yang H.(1999) Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal: 66-79.
[86] Yang H., Zhang L.(2001a) The joint distribution of surplus immediately before ruin and the deficit at ruin under interest force. North American Actuarial Journal, 5 : 92-103.
[87] Yang H. Zhang L.(2001b) On the distribution of the surplus immediately after ruin under interest force. Insurance: Mathematics and Economics, 29: 247-256.