基于两个独立的指数随机变量之和的相依风险模型的破产问题研究
Research on the Ruin Problem of Dependent Risk Model Based on the Sum of Two Independent Exponential Random Variables
摘要: 在本文中我们考虑了经典复合泊松模型的一个推广，该模型的累计索赔额过程的增量是独立的。其中索赔间隔时间的分布是两个独立的指数随机变量之和，在本文中我们考虑了索赔时间与后续索赔规模之间的一种特殊的依赖结构，导出了Gerber-Shiu惩罚函数的积分微分方程，利用Rouché定理研究了林德伯格等式的根，导出了惩罚函数的拉普拉斯变换及其满足的瑕疵更新方程。最后在索赔时间间隔服从两个独立的指数随机变量之和分布时给出破产概率的解析解，对理论结果做数值分析，对不同相依参数下的破产概率进行对比。
Abstract: In this paper, we consider an extension of the classical compound Poisson model where the incre-mental cumulative claims process is independent. The distribution of claim interval follows the sum of two independent random variables. In this paper, we consider a special dependence structure between claim time and subsequent claim scale, derive the Gerber-Shiu penalty function integral differential equation, and use Rouché’s theorem to study the roots of Lundberg equation. The La-place transform of penalty function and its defect renewal equation are derived. Finally, the analytic solution of ruin probability is given when the claim interval follows the sum distribution of two in-dependent exponential random variables. The theoretical results are analyzed numerically, the ru-in probabilities under different dependent parameters are compared.
文章引用：王婧璇. 基于两个独立的指数随机变量之和的相依风险模型的破产问题研究[J]. 应用数学进展, 2023, 12(7): 3447-3462. https://doi.org/10.12677/AAM.2023.127342

参考文献

[1] Sparre Andersen, E. (1957) On the Collective Theory of Risk in the Case of Contagion between the Claims. Transactions of the XV International Congress of Actuaries, 2, 219-229.
[2] Dickson, D.C.M. (1992) On the Distribution of the Surplus Prior to Ruin. Insurance: Mathematics and Economics, 11, 191-207. [Google Scholar] [CrossRef] 
[3] Dickson, D.C.M. (1993) On the Distribution of the Claim Causing Ruin. Insurance: Mathematics and Economics, 12, 143-254. [Google Scholar] [CrossRef] 
[4] Gerber, H.U. and Shiu, E.S.W. (1997) The Joint Distribution of the Time of Ruin, the Surplus Immediately before Ruin and the Defect at Ruin. Insurance: Mathematics and Econom-ics, 22, 129-137. [Google Scholar] [CrossRef] 
[5] Dickson, D.C.M. and Hipp, C. (1998) Ruin Probabilities for Erlang(2) Risk Process. Insurance: Mathematics and Economics, 22, 251-262. [Google Scholar] [CrossRef] 
[6] Rolski, T., Schmidli, H., Schmidt, V., et al. (2008) Stochastic Processes for Insurance and Finance. John Wiley & Sons Ltd. [Google Scholar] [CrossRef] 
[7] Ger-ber, H.U. and Shiu, E.S.W. (1998) On the Time Value of Ruin. North American Actuarial Journal, 2, 48-78. [Google Scholar] [CrossRef] 
[8] Dickson, D.C.M. and Hipp, C. (2001) On the Time to Ruin for Erlang(2) Risk Process. Insurance: Mathematics and Economic, 229, 333-344. [Google Scholar] [CrossRef] 
[9] Gerber, H.U. and Shiu, E.S.W. (2005) The Time Value of Ruin in a Sparre Andersen Model. North American Actuarial Journal, 9, 49-69. [Google Scholar] [CrossRef] 
[10] Li, S. and Garrido, J. (2004) On Ruin for the Erlang(n) Risk Process. Insurance: Mathematics and Economic, 34, 391-408. [Google Scholar] [CrossRef] 
[11] Albrecher, H. and Boxma, O.J. (2004) A Ruin Model with Dependence between Claim Sizes and Claim Intervals. Insurance: Mathematics and Economics, 35, 245-254. [Google Scholar] [CrossRef] 
[12] Albrecher, H. and Teugels, J. (2006) Exponential Behavior in the Presence of Dependence in Risk Theory. Journal of Applied Probability, 43, 257-273. [Google Scholar] [CrossRef] 
[13] Zhang, L. and Liu, H. (2020) On a Discrete-Time Risk Model with Time-Dependent Claims and Impulsive Dividend Payments. Scandinavian Actuarial Journal, 2020, 736-753. [Google Scholar] [CrossRef] 

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