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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (2): 121-130.doi: 10.3969/j.issn.1005-3085.2016.02.002

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变保费率复合Poisson-Geometric过程风险\\[3mm]模型的Gerber-Shiu折现惩罚函数

贺丽娟1,  王成勇2,  张  锴1   

  1. 1- 文华学院基础学部，武汉  430074
    2- 湖北文理学院数学与计算机科学学院，襄阳 441053
  • 收稿日期:2015-05-14 接受日期:2015-11-16 出版日期:2016-04-15 发布日期:2016-06-15
  • 通讯作者: 张 锴 E-mail: 23207882@qq.com
  • 基金资助:
    国家自然科学基金 (71371066).

Gerber-Shiu Discounted Penalty Function for Compound Poisson-Geometric Risk Model with Variable Premium Rate

HE Li-juan1,  WANG Cheng-yong2,  ZHANG Kai1   

  1. 1- Department of Mathematics, Wenhua College, Wuhan 430074
    2- College of Mathematics & Computer Science, Hubei University of Arts of Science, Xiangyang 441053
  • Received:2015-05-14 Accepted:2015-11-16 Online:2016-04-15 Published:2016-06-15
  • Contact: K. Zhang.E-mail address: 23207882@qq.com
  • Supported by:
    The National Natural Science Foundation of China (71371066).

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摘要： 本文研究了当保费率随时间变化时的复合Poisson-Geometric过程的风险模型．通过无穷小方法，得到了该模型的Gerber-Shiu折现惩罚函数所满足的更新方程．在此基础上，推导出破产概率，破产前瞬时盈余，以及破产时刻赤字分布满足的更新方程．特别地，当个体索赔服从指数分布时，通过求解微分方程，得到了该模型的破产概率的显式表达式和所满足的不等式．最后通过数值模拟和算例分析，提出了保险公司的赔付政策和保费政策对自身风险的影响．

关键词: 变保费率, 复合Poisson-Geometric过程, Gerber-Shiu折现惩罚函数, 破产概率

Abstract:

In this paper, we consider a new risk model of compound Poisson-Geometric process which assumes that the insurance company receives the premium with a differentiable rate. By applying the differential argument method, a defective renewal equation of Gerber-Shiu discounted penalty function is obtained. Based on the results, the defective renewal equation of the ruin probability, the moments of the surplus immediately prior to ruin and the deficit at ruin have been deduced. By solving the differential equation, the inequality which the ruin probability satisfied have been obtained when the claim variable random belongs to the exponential distribution. Moreover, numerical analysis of the distribution is presented and some examples are given. Finally, we conclude that the impacts of the adjustment policy and the premium policy to the insurance company.

Key words: variable premium rate, compound Poisson-Geometric process, Gerber-Shiu discounted penalty function, ruin probability

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