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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 ›› 2020, Vol. 37 ›› Issue (1): 89-106.doi: 10.3969/j.issn.1005-3085.2020.01.008

• • 上一篇    下一篇

带延迟索赔和随机收入的离散风险模型的Gerber-Shiu分析（英）

黄  娅1,  刘  娟2,  周杰明2,  邓迎春2   

  1. 1- 湖南师范大学商学院，长沙  410081
    2- 湖南师范大学数学与统计学院，计算与随机数学教育部重点实验室，长沙 410081
  • 收稿日期:2017-11-21 接受日期:2018-06-19 出版日期:2020-02-15 发布日期:2020-04-15
  • 通讯作者: 邓迎春 E-mail: dengyc@hunnu.edu.cn
  • 基金资助:
    湖南省哲学社会科学基金(17YBA290)；湖南省教育厅科学研究项目(17K057; 17C1001).

Gerber-Shiu Analysis for a Discrete Risk Model with Delayed Claims and Random Incomes

HUANG Ya1,  LIU Juan2,  ZHOU Jie-ming2,  DENG Ying-chun2   

  1. 1- School of Business, Hunan Normal University, Changsha 410081
    2- Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081
  • Received:2017-11-21 Accepted:2018-06-19 Online:2020-02-15 Published:2020-04-15
  • Contact: Y. Deng. E-mail address: dengyc@hunnu.edu.cn
  • Supported by:
    The Philosophy and Social Science Fund of Hunan Province (17YBA290); the Scientific Research Fund of Hunan Provincial Department of Education (17K057; 17C1001).

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摘要： 破产理论是保险数学中的重要问题，它可以为保险公司决策者提供一个非常有用的早期风险预警手段．本文研究了一个带潜在延迟索赔和随机保费收入的复合二项风险模型．利用矩母函数的技巧，得到了 Gerber-Shiu 期望折罚函数的递推公式．特别地，还得到了贴现因子为 1 的特殊情形下的 Gerber-Shiu 期望折罚函数的解析表达式．最后还得到了实际应用中的一些重要的破产特征量，包括破产概率，破产时赤字的密度函数，破产前盈余与破产时赤字的联合密度函数，以及导致破产的索赔密度函数等．

关键词: 复合二项风险模型, Gerber-Shiu 期望折罚函数, 延迟索赔, 随机保费, 递推公式

Abstract: Ruin theory is the mainly contents of insurance mathematics, as it can supply a very useful early-warning measure for the risk of the insurance company. In this paper, we study a risk model with potentially delayed claims and random premium incomes within the framework of the compound binomial model. Using the technique of generating functions, we derive a recursive formula for the Gerber-Shiu expected discounted penalty function. Specifically, an explicit formula is obtained for the discount-free case. As applications, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

Key words: compound binomial risk model, Gerber-Shiu expected discounted penalty function, delayed claims, random premium, recursive formula

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