索赔额服从混合指数分布的一类期望折现罚金函数
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摘要
基于经典风险模型,针对指数索赔间隔和混合指数索赔额的情况,研究关于实质破产的期望折现罚金函数.首先,利用全概率公式得到期望折现罚金函数满足的积分微分方程;然后,在索赔额为混合指数分布的情况下推导出期望折现罚金函数满足的微分方程,进而针对常数破产率函数,得到期望折现罚金函数的具体表达式.
Based on the classical risk model,the expected discounted penalty function about bankruptcy is studied with exponentially inter-claim times and mixed exponentially claim size. First,the integro-differential equations that satisfy discounted penalty function are obtained by total probability formula. Then,the differential equations for the discount penalty function are obtained when the claim size is mixed exponentially distribution,and for constant bankruptcy rate function,the explicit formula of the discount penalty function is deduced.
Based on the classical risk model,the expected discounted penalty function about bankruptcy is studied with exponentially inter-claim times and mixed exponentially claim size. First,the integro-differential equations that satisfy discounted penalty function are obtained by total probability formula. Then,the differential equations for the discount penalty function are obtained when the claim size is mixed exponentially distribution,and for constant bankruptcy rate function,the explicit formula of the discount penalty function is deduced.
引文
[1]ALBRECHER H,CHEUNG C K,THONHAUSER S.Randomized observation periods for the compound Possion risk model:The discounted penalty function[J].Scandinavian Actuarial Journal,2013,13(6):424-452.
[2]GERBER H U,SHIU E S W,YANG H L.The Omega model:From bankruptcy to occupation time in the red[J].European Actuarial Journal,2012,2(2):259-272.
[3]GERBER H U,SHIU E S W.On the time value of ruin[J].North American Actuarial Journal,1998,2(1):48-72.
[4]SONG M,MENG Q B,WU R,et al.The Gerber-Shiu discounted penalty function in the risk process with phase-type interclaim times[J].Applied Mathematics and Computation,2010,216(2):523-531.
[5]DONG Y H,ZHANG H J.Ruin probability in risk model with two Poisson processes by diffusion[J].Mathematical Theory and Application,2003,23(1):98-101.
[6]GERBER H U,SHIU E S W.The time value of ruin in a Sparre Andersen Model[J].North American Actuarial Journal,2005,9(2):49-69.
[7]ALBRECHER H,GERBER H U,SHIU E S W.The optimal dividend barrier in the Gamma-Omega model[J].European Actuarial Journal,2011,1(1):43-56.
[8]ALBRECHER H,LAUTSCHAM V.From ruin to bankruptcy for compound Poisson surplus processes[J].Astin Bulletin,2013,43(2):213-243.
[9]李野默,王秀莲.复合泊松风险模型中观察间隔为混合指数分布的贴现罚金函数[J].天津师范大学学报(自然科学版),2015,35(2):17-20.LI Y M,WANG X L.Discounted penalty function of compound Poisson risk model when observation interval being mixed exponential distribution[J].Journal of Tianjin Normal University(Natural Science Edition),2015,35(2):17-20(in Chinese).
[2]GERBER H U,SHIU E S W,YANG H L.The Omega model:From bankruptcy to occupation time in the red[J].European Actuarial Journal,2012,2(2):259-272.
[3]GERBER H U,SHIU E S W.On the time value of ruin[J].North American Actuarial Journal,1998,2(1):48-72.
[4]SONG M,MENG Q B,WU R,et al.The Gerber-Shiu discounted penalty function in the risk process with phase-type interclaim times[J].Applied Mathematics and Computation,2010,216(2):523-531.
[5]DONG Y H,ZHANG H J.Ruin probability in risk model with two Poisson processes by diffusion[J].Mathematical Theory and Application,2003,23(1):98-101.
[6]GERBER H U,SHIU E S W.The time value of ruin in a Sparre Andersen Model[J].North American Actuarial Journal,2005,9(2):49-69.
[7]ALBRECHER H,GERBER H U,SHIU E S W.The optimal dividend barrier in the Gamma-Omega model[J].European Actuarial Journal,2011,1(1):43-56.
[8]ALBRECHER H,LAUTSCHAM V.From ruin to bankruptcy for compound Poisson surplus processes[J].Astin Bulletin,2013,43(2):213-243.
[9]李野默,王秀莲.复合泊松风险模型中观察间隔为混合指数分布的贴现罚金函数[J].天津师范大学学报(自然科学版),2015,35(2):17-20.LI Y M,WANG X L.Discounted penalty function of compound Poisson risk model when observation interval being mixed exponential distribution[J].Journal of Tianjin Normal University(Natural Science Edition),2015,35(2):17-20(in Chinese).