几类阈红利边界策略下Gerber-Shiu罚金折现函数研究
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摘要
破产,是指保险人在拥有一定初始资产的前提下,经过一段时间的经营,盈余第一次变为负值的情况,只是一个数学概念,并不一定意味着保险公司就此倒闭。在现代破产理论中,普遍关注的三个重要指标是：破产概率、破产时间、破产前瞬时盈余和破产赤字之间的关系。Hans U.Gerber和Elias S.W.Shiu构造了著名的期望折现罚金函数,也叫做Gerber-Shiu罚金折现函数。该函数引入破产前瞬时盈余和破产赤字两个指标,非常方便的刻画了破产概率、破产事件的Laplace变换以及破产前瞬时盈余与破产赤字的联合密度函数间关系。保险风险模型中的分红策略最初由De Finetti提出,旨在更实际的反应一个保险投资组合的现金流。之后,与盈余有关的两种分红策略引起了我们的重视：一种是常数值红利边界风险模型又称为完全分红模型,当盈余低于一个常值时,没有红利发给股东或者投保人,然而,盈余一旦高于此边界,则超过的全部盈余都作为红利发给股东,Gerber最初研究了这种策略。另一种是阈红利边界策略,这种风险模型规定,当盈余高于边界时,则红利以低于保费收入的部分发给股东或者投保人,这个策略首先是Gerber、Buhlmann提出,之后在常数边界和依赖于时间的线性边界下,许多学者作了许多工作。本文引入上述两种分红策略以及经典风险模型为基础,同时引入相依风险和带扰动的经典风险模型,然后得出上述对两种分红策略有关结论进行推广,首先引入常数红利边界下阈红利策略,并给出该模型下Gerber-Shiu罚金折现函数,同时给出了线性边界下Gerber-Shiu罚金折现函数结果；引入线性边界下阈红利策略和相依风险模型同时给出满足这两种情况的Gerber-Shiu罚金折现函数；引入带扰动风险模型,得出这种情况下的生存概率、红利付款的期望现值、Gerber-Shiu罚金折现函数等结果。
Ruin theory means that insurers have some initial premise of the assets, after a period of operation, Surplus for the first time into a negative situation, only a mathematical concept, this failure does not necessarily mean the insurance company ruin. In modern Ruin theory, three important indicators of general concern are:the time of Ruin, the deficit at ruin, the surplus immediately before the time of ruin.
Hans U. Gerber and EliasS.W.Shiu constructed the famous expected dis-counted penalty function, also called Gerber-Shiu discounted penalty function. The instantaneous profit function is the deficit at ruin and the surplus immediately prior to ruin two indicators, Very convenient to describe the probability of Ruin, ruin events prior to the insolvency of the Laplace transform as well as the deficit at ruin and the surplus immediately prior to ruin between the joint density function.
Insurance risk model in the initial dividend strategy proposed by the De Finetti, more realistic response to an insurance portfolio cash flow. After two kinds of dividend strategy attracted our attention:one is the constant value Dividend risk model is also known as total dividend model, when the surplus under a constant value, no dividends to shareholders or the insurant, However, once surplus above this boundary, all the surplus over all as a dividend to shareholders, Gerber initial study of this strategy. The other is the threshold dividend strategy, the provisions of this risk model, when the surplus above the boundary, then the dividend less than the premium income to shareholders or policyholders part of this strategy first Gerber, Buhlmann presented, followed by the boundary and depends on the time constant of the linear boundary, many scholars made a lot of work. This dividend strategy and the introduction of the two classical risk model, while the introduction of dependent risks and the classical risk model with a disturbance, and then draw the relevant conclusions of the two dividend strategy to promote, first introduced under the threshold constant Dividend dividend strategy, given the model Gerber-Shiu discounted penalty function, the linear Dividend Barrier is given under the Gerber-Shiu discounted penalty function results; the introduction of linear Dividend Barrier and dependent under the threshold dividend strategy risk model is given to satisfy these two situation Gerber-Shiu discounted penalty function; the introduction of the risk model perturbed obtained the probability of survival under such circumstances, the present value of expected dividend payments, Gerber-Shiu discounted penalty function and other results.
Ruin theory means that insurers have some initial premise of the assets, after a period of operation, Surplus for the first time into a negative situation, only a mathematical concept, this failure does not necessarily mean the insurance company ruin. In modern Ruin theory, three important indicators of general concern are:the time of Ruin, the deficit at ruin, the surplus immediately before the time of ruin.
Hans U. Gerber and EliasS.W.Shiu constructed the famous expected dis-counted penalty function, also called Gerber-Shiu discounted penalty function. The instantaneous profit function is the deficit at ruin and the surplus immediately prior to ruin two indicators, Very convenient to describe the probability of Ruin, ruin events prior to the insolvency of the Laplace transform as well as the deficit at ruin and the surplus immediately prior to ruin between the joint density function.
Insurance risk model in the initial dividend strategy proposed by the De Finetti, more realistic response to an insurance portfolio cash flow. After two kinds of dividend strategy attracted our attention:one is the constant value Dividend risk model is also known as total dividend model, when the surplus under a constant value, no dividends to shareholders or the insurant, However, once surplus above this boundary, all the surplus over all as a dividend to shareholders, Gerber initial study of this strategy. The other is the threshold dividend strategy, the provisions of this risk model, when the surplus above the boundary, then the dividend less than the premium income to shareholders or policyholders part of this strategy first Gerber, Buhlmann presented, followed by the boundary and depends on the time constant of the linear boundary, many scholars made a lot of work. This dividend strategy and the introduction of the two classical risk model, while the introduction of dependent risks and the classical risk model with a disturbance, and then draw the relevant conclusions of the two dividend strategy to promote, first introduced under the threshold constant Dividend dividend strategy, given the model Gerber-Shiu discounted penalty function, the linear Dividend Barrier is given under the Gerber-Shiu discounted penalty function results; the introduction of linear Dividend Barrier and dependent under the threshold dividend strategy risk model is given to satisfy these two situation Gerber-Shiu discounted penalty function; the introduction of the risk model perturbed obtained the probability of survival under such circumstances, the present value of expected dividend payments, Gerber-Shiu discounted penalty function and other results.
引文
[1]Albrecher, H.,Box ma,O.J.,2004.A ruin model with dependence be-tween claim size and claim intervals, [J] Insurance:Mathematics and Economics,35:245-254
[2]Albrecher,H,et.al..2005,On the Distribution of Dividend payments and the Discounted penalty Function in a risk Model with linear Dividend Bar-rier,Scandinavian Actuarical Journal,2:103-126
[3]Asmussen,S.,Taksar,M..Controlled diffusion models for optimal dividend pay-out.Insurance:Mathematics and Economics20(1997)1-15
[4]BjΦrn Sundt.Recursive evaluation of aggregate claims distributions.Insurance: Mathematics and Economics 30 (2002) 297-322
[5]Boudreault M,Cossetteh H.,Landriault D.,Marceau E.2006.On the risk model with dependence between interclaim arrivals and claim size,[J]Scandinavian Actuarial Journal.265-285
[6]Cary Chi-Liang Tsai.On the discounted distribution functions of the surplus process perturbed by diffusion.Insurance:Mathematics and Economics 28 (2001)401-419
[7]Cary Chi-Liang Tsai, Gordon E. Willmot.A generalized defective renewal equation for the surplus process perturbed by diffusion.Insurance:Mathemat-ics and Economics 30 (2002) 51-66
[8]Cary Chi-Liang Tsai, Gordon E. Willmot.On the moments of the surplus pro-cess perturbed by diffusion.Insurance:Mathematics and Economics 31 (2002) 327-350
[9]Cary Chi-Liang Tsai.On the expectations of the present values of the time of ruin perturbed by diffusion.Insurance:Mathematics and Economics 32 (2003) 413-429
[10]Cary Chi-Liang Tsai.On the stop-loss transform and order for the surplus pro-cess perturbed by diffusion.Insurance Mathematics and Economics 39 (2006) 151-170
[11]David Landriault.Constant dividend barrier in a risk model with interclaim-dependent claim.Insurance:Mathematics and Economics 42 (2008) 31-38
[12]David Landriault, Gordon Willmot.On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time dis-tribution.Insurance:Mathematics and Economics 42 (2008) 600-608
[13]David C.M. Dickson, Christian Hipp.On the time to ruin for Erlang(2) risk processes.Insurance:Mathematics and Economics 29 (2001) 333-344
[14]Franqois Dufresne,Hans U. Gerber.Risk theory for the compound Poisson pro-cess that is perturbed by diffusion.Insurance:Mathematics and Economics 10 (1991)51-59
[15]Gerber,H.U;Elias,S.W.Shiu.1998.on the time value of ruin[J],North American Actuarial Jornal,21:48-78
[16]Guojing Wang, Rong Wu.Some distributions for classical risk process that is perturbed by diffusion.Insurance:Mathematics and Economics 26 (2000) 15-24
[17]Guo-jing Wang, Rong Wu.Some Results for Classical Risk Process with Stochastic Return on Investments.Acta Mathematicae Applicatae Sinica, En-glish Series Vol.18, No.4 (2002) 685-692
[18]Guojing Wanga, Rong Wu.The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest.Insurance: Mathematics and Economics 42 (2008) 59-64.
[19]
[20]H Albrecher, Onno J. Boxma.A ruin model with dependence between claim sizes and claim intervals.Insurance:Mathematics and Economics 35 (2004) 245-254
[21]H Albrecher, J Hartinger,RF Tichy,On the Distribution of Dividend Payments and the Discounted Penalty Function in a Risk Model with Linear Dividend Barrier.Scandinavian Actuarial Journal.2005-gec.unil.ch
[22]Hans U. Gerber, Bruno Landry.On the discounted penalty at ruin in a jump-diffusion and the perpetual put option.Insurance:Mathematics and Economics 22(1998)263-276
[23]Hans U. Gerber,Elias S. W. Shiu.Optimal dividends:analtsis with brownian motion.North american actuarial journal,Volume 8,Number 1
[24]Heli Gao, Chuancun Yin.The perturbed Sparre Andersen model with a thresh-old dividend strategy.Journal of Computational and Applied Mathematics 220 (2008)394-408
[25]H. Schmidli.Crame'r-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion.Insurance:Mathematics and Economics 16 (1995) 135-149
[26]H. Schmidli.An. Extension to the renewal theorem and an application to risk theory. The Annals of Applied Probability 1997, Vol.7, No.1,121-133
[27]Jeanblanc-picque',M.,Shiryaev,A.A.Optimization of flow of divi-dends.Russian Mathematical Surveys(1995)20,257-277
[28]Joykrishna Sarkar, Arusharka Sen.Weak convergence approach to compound Poisson risk processes perturbed by diffusion.Insurance:Mathematics and Economics 36 (2005) 421-432
[29]Kam C. Yuena, Guojing Wang, Wai K. Li.The Gerber-Shiu expected dis-counted penalty function for risk processes with interest and a constant divi-dend barrier.Insurance:Mathematics and Economics 40 (2007) 104-112
[30]Konstantin A. Borovkov, David C.M. Dickson.On the ruin time distribution for a Sparre Andersen process with exponential claim sizes.Insurance:Math-ematics and Economics 42 (2008) 1104-1108
[31]Landriault David.2008.Constant dividend barrier in a risk model with interclaim-dependent claim sizes,[J].Insurance:Mathematics and Economics.42:31-38
[32]Lin X.S,Willmot G.E.,Steve D,2003.The classical model with a con-stant dievidend barrier:anlysis of the Gerber-Shiu discounted penalty func-tion.Insurance:Mathematics and Economics33:551-566
[33]M. Boudreault et al.On a risk model with dependence between interclaim ar-rivals and claim sizes.Scandinavian Actuarial Journal,2006,5,265-285
[34]Manuel Morales.On the expected discounted penalty function for a perturbed risk process driven by a subordinator.Insurance:Mathematics and Economics 40(2007)293-301
[35]Maria de Lourdes Centeno.Dependent risks and excess of loss reinsur-ance.Insurance:Mathematics and Economics 37 (2005) 229-238
[36]Ning Wan.Dividend payments with a threshold strategy in the compound Pois-son risk model perturbed by diffusion.Insurance:Mathematics and Economics 40 (2007) 509-523
[37]Rob Kaas,Marc Goovaerts,Jan Dhaene,Michel Denuit.Modern Actuarial Risk Theory.Kluwer Academic Publishers.2002
[38]Schmidli, H., On the Gerber-Shiu function and change of measure.Insurance:Mathematics and Economics (2009), doi:10.1016/j.insmatheco.2009.04.004
[39]Shuanming Li, Jos'e Garrido.On a class of renewal risk models with a constant dividend barrier.Insurance:Mathematics and Economics 35 (2004) 691-701
[40]Soohan Ahn, Andrei L. Badescu.On the analysis of the Gerber-Shiu dis-counted penalty function for risk processes with Markovian arrivals.Insurance: Mathematics and Economics 41 (2007) 234-249
[41]Stathis Chadjiconstantinidis, Georgios Pitselis.Further improved recursions for a class of compound Poisson distributions.Insurance:Mathematics and Economics.
[42]SΦren Asmussen, Michael Taksar.Controlled diffusion models for optimal dividend pay-out.Insurance:Mathematics and Economics 20 (1997) 1-15
[43]Thomas Siegl, Robert F. Tichy.A process with stochastic claim frequency and a linear dividend barrier.Insurance:Mathematics and Economics 24 (1999) 51-65
[44]X. Sheldon Lin, Gordon E. Willmot.Analysis of a defective renewal equa-tion arising in ruin theory.Insurance:Mathematics and Economics 25 (1999) 63-84
[45]X. Sheldon Lin, Kristina P. Sendova.The compound Poisson risk model with multiple thresholds.Insurance:Mathematics and Economics 42 (2008) 617-627
[46]X. Sheldon Lin, Gordon E. Willmot, Steve Drekic.The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function.Insurance:Mathematics and Economics 33 (2003) 551-566
[47]X. Sheldon Lin, Kristina P. Pavlova.The compound Poisson risk model with a threshold dividend strategy.Insurance:Mathematics and Economics 38 (2006) 57-80
[48]Zhen-hua Bao, Zhong-xing Ye.The Gerber-Shiu discounted penalty function in the delayed renewal risk process with random income. Applied Mathematics and Computation 184 (2007) 857-863
[49]李波,吴荣.跳扩散对偶模型在带壁分红策略下的分红函数.应用数学和力学,第29卷第9期1124-1134
[50]花兆秀,牛明飞.在索赔额相依风险模型中的阈值分红策略.山东大学学报(理学版)2008.43：91-96
[51]张燕,田铮,刘向增.阈红利边界下理赔时间间隔与理赔额相依的风险模型.数理统计与管理.2008.Jul：730-739
[52]宗昭军,胡锋,,元春梅.具有线性红利界限的破产理论.工程数学学报.第23卷第2期(2006)：319-323
[53]汉斯,U.盖伯著,成世学,严颖译,1997,数学风险论导引,世界图书出版公司北京公司
[54][美]sheldon劳斯.随机过程[M].北京：中国统计出版社,2005.
[55]杨莉等.线性边界下两步保费率风险模型的Gerber-Shiu罚金函数.数学的实践与认识,第37卷第11期.
[2]Albrecher,H,et.al..2005,On the Distribution of Dividend payments and the Discounted penalty Function in a risk Model with linear Dividend Bar-rier,Scandinavian Actuarical Journal,2:103-126
[3]Asmussen,S.,Taksar,M..Controlled diffusion models for optimal dividend pay-out.Insurance:Mathematics and Economics20(1997)1-15
[4]BjΦrn Sundt.Recursive evaluation of aggregate claims distributions.Insurance: Mathematics and Economics 30 (2002) 297-322
[5]Boudreault M,Cossetteh H.,Landriault D.,Marceau E.2006.On the risk model with dependence between interclaim arrivals and claim size,[J]Scandinavian Actuarial Journal.265-285
[6]Cary Chi-Liang Tsai.On the discounted distribution functions of the surplus process perturbed by diffusion.Insurance:Mathematics and Economics 28 (2001)401-419
[7]Cary Chi-Liang Tsai, Gordon E. Willmot.A generalized defective renewal equation for the surplus process perturbed by diffusion.Insurance:Mathemat-ics and Economics 30 (2002) 51-66
[8]Cary Chi-Liang Tsai, Gordon E. Willmot.On the moments of the surplus pro-cess perturbed by diffusion.Insurance:Mathematics and Economics 31 (2002) 327-350
[9]Cary Chi-Liang Tsai.On the expectations of the present values of the time of ruin perturbed by diffusion.Insurance:Mathematics and Economics 32 (2003) 413-429
[10]Cary Chi-Liang Tsai.On the stop-loss transform and order for the surplus pro-cess perturbed by diffusion.Insurance Mathematics and Economics 39 (2006) 151-170
[11]David Landriault.Constant dividend barrier in a risk model with interclaim-dependent claim.Insurance:Mathematics and Economics 42 (2008) 31-38
[12]David Landriault, Gordon Willmot.On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time dis-tribution.Insurance:Mathematics and Economics 42 (2008) 600-608
[13]David C.M. Dickson, Christian Hipp.On the time to ruin for Erlang(2) risk processes.Insurance:Mathematics and Economics 29 (2001) 333-344
[14]Franqois Dufresne,Hans U. Gerber.Risk theory for the compound Poisson pro-cess that is perturbed by diffusion.Insurance:Mathematics and Economics 10 (1991)51-59
[15]Gerber,H.U;Elias,S.W.Shiu.1998.on the time value of ruin[J],North American Actuarial Jornal,21:48-78
[16]Guojing Wang, Rong Wu.Some distributions for classical risk process that is perturbed by diffusion.Insurance:Mathematics and Economics 26 (2000) 15-24
[17]Guo-jing Wang, Rong Wu.Some Results for Classical Risk Process with Stochastic Return on Investments.Acta Mathematicae Applicatae Sinica, En-glish Series Vol.18, No.4 (2002) 685-692
[18]Guojing Wanga, Rong Wu.The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest.Insurance: Mathematics and Economics 42 (2008) 59-64.
[19]
[20]H Albrecher, Onno J. Boxma.A ruin model with dependence between claim sizes and claim intervals.Insurance:Mathematics and Economics 35 (2004) 245-254
[21]H Albrecher, J Hartinger,RF Tichy,On the Distribution of Dividend Payments and the Discounted Penalty Function in a Risk Model with Linear Dividend Barrier.Scandinavian Actuarial Journal.2005-gec.unil.ch
[22]Hans U. Gerber, Bruno Landry.On the discounted penalty at ruin in a jump-diffusion and the perpetual put option.Insurance:Mathematics and Economics 22(1998)263-276
[23]Hans U. Gerber,Elias S. W. Shiu.Optimal dividends:analtsis with brownian motion.North american actuarial journal,Volume 8,Number 1
[24]Heli Gao, Chuancun Yin.The perturbed Sparre Andersen model with a thresh-old dividend strategy.Journal of Computational and Applied Mathematics 220 (2008)394-408
[25]H. Schmidli.Crame'r-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion.Insurance:Mathematics and Economics 16 (1995) 135-149
[26]H. Schmidli.An. Extension to the renewal theorem and an application to risk theory. The Annals of Applied Probability 1997, Vol.7, No.1,121-133
[27]Jeanblanc-picque',M.,Shiryaev,A.A.Optimization of flow of divi-dends.Russian Mathematical Surveys(1995)20,257-277
[28]Joykrishna Sarkar, Arusharka Sen.Weak convergence approach to compound Poisson risk processes perturbed by diffusion.Insurance:Mathematics and Economics 36 (2005) 421-432
[29]Kam C. Yuena, Guojing Wang, Wai K. Li.The Gerber-Shiu expected dis-counted penalty function for risk processes with interest and a constant divi-dend barrier.Insurance:Mathematics and Economics 40 (2007) 104-112
[30]Konstantin A. Borovkov, David C.M. Dickson.On the ruin time distribution for a Sparre Andersen process with exponential claim sizes.Insurance:Math-ematics and Economics 42 (2008) 1104-1108
[31]Landriault David.2008.Constant dividend barrier in a risk model with interclaim-dependent claim sizes,[J].Insurance:Mathematics and Economics.42:31-38
[32]Lin X.S,Willmot G.E.,Steve D,2003.The classical model with a con-stant dievidend barrier:anlysis of the Gerber-Shiu discounted penalty func-tion.Insurance:Mathematics and Economics33:551-566
[33]M. Boudreault et al.On a risk model with dependence between interclaim ar-rivals and claim sizes.Scandinavian Actuarial Journal,2006,5,265-285
[34]Manuel Morales.On the expected discounted penalty function for a perturbed risk process driven by a subordinator.Insurance:Mathematics and Economics 40(2007)293-301
[35]Maria de Lourdes Centeno.Dependent risks and excess of loss reinsur-ance.Insurance:Mathematics and Economics 37 (2005) 229-238
[36]Ning Wan.Dividend payments with a threshold strategy in the compound Pois-son risk model perturbed by diffusion.Insurance:Mathematics and Economics 40 (2007) 509-523
[37]Rob Kaas,Marc Goovaerts,Jan Dhaene,Michel Denuit.Modern Actuarial Risk Theory.Kluwer Academic Publishers.2002
[38]Schmidli, H., On the Gerber-Shiu function and change of measure.Insurance:Mathematics and Economics (2009), doi:10.1016/j.insmatheco.2009.04.004
[39]Shuanming Li, Jos'e Garrido.On a class of renewal risk models with a constant dividend barrier.Insurance:Mathematics and Economics 35 (2004) 691-701
[40]Soohan Ahn, Andrei L. Badescu.On the analysis of the Gerber-Shiu dis-counted penalty function for risk processes with Markovian arrivals.Insurance: Mathematics and Economics 41 (2007) 234-249
[41]Stathis Chadjiconstantinidis, Georgios Pitselis.Further improved recursions for a class of compound Poisson distributions.Insurance:Mathematics and Economics.
[42]SΦren Asmussen, Michael Taksar.Controlled diffusion models for optimal dividend pay-out.Insurance:Mathematics and Economics 20 (1997) 1-15
[43]Thomas Siegl, Robert F. Tichy.A process with stochastic claim frequency and a linear dividend barrier.Insurance:Mathematics and Economics 24 (1999) 51-65
[44]X. Sheldon Lin, Gordon E. Willmot.Analysis of a defective renewal equa-tion arising in ruin theory.Insurance:Mathematics and Economics 25 (1999) 63-84
[45]X. Sheldon Lin, Kristina P. Sendova.The compound Poisson risk model with multiple thresholds.Insurance:Mathematics and Economics 42 (2008) 617-627
[46]X. Sheldon Lin, Gordon E. Willmot, Steve Drekic.The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function.Insurance:Mathematics and Economics 33 (2003) 551-566
[47]X. Sheldon Lin, Kristina P. Pavlova.The compound Poisson risk model with a threshold dividend strategy.Insurance:Mathematics and Economics 38 (2006) 57-80
[48]Zhen-hua Bao, Zhong-xing Ye.The Gerber-Shiu discounted penalty function in the delayed renewal risk process with random income. Applied Mathematics and Computation 184 (2007) 857-863
[49]李波,吴荣.跳扩散对偶模型在带壁分红策略下的分红函数.应用数学和力学,第29卷第9期1124-1134
[50]花兆秀,牛明飞.在索赔额相依风险模型中的阈值分红策略.山东大学学报(理学版)2008.43：91-96
[51]张燕,田铮,刘向增.阈红利边界下理赔时间间隔与理赔额相依的风险模型.数理统计与管理.2008.Jul：730-739
[52]宗昭军,胡锋,,元春梅.具有线性红利界限的破产理论.工程数学学报.第23卷第2期(2006)：319-323
[53]汉斯,U.盖伯著,成世学,严颖译,1997,数学风险论导引,世界图书出版公司北京公司
[54][美]sheldon劳斯.随机过程[M].北京：中国统计出版社,2005.
[55]杨莉等.线性边界下两步保费率风险模型的Gerber-Shiu罚金函数.数学的实践与认识,第37卷第11期.