The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function
The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function
复制标题
DOI：
10.1007/s11009-007-9050-6
复制
发表时间：
2009-09
影响因子：
0.9
通讯作者：
Jun Cai;Runhuan Feng;G. Willmot
中科院分区：
文献类型：
--
作者：
Jun Cai;Runhuan Feng;G. Willmot
我们修改的复合泊松盈余模型的保险公司，包括流动准备金和盈余的利息。当保险公司的盈余低于固定水平时，盈余将作为流动准备金保留，不赚取利息。当盈余达到这个水平时，超过这个水平的盈余将按固定利率收取利息。当风险水平趋于无穷大时，修正后的模型就退化为经典的复合泊松风险模型。如果水平设置为零，则修改后的模型变为带利率的复合泊松风险模型。利用Gerber-Shiu函数研究了修正的复合Poisson盈余模型中的破产概率及其他与破产相关的量，讨论了利率和流动准备金对破产概率、破产赤字及其他破产量的影响.首先，我们推导出Gerber-Shiu函数的积分微分方程组。通过求解方程组，我们得到了Gerber-Shiu函数的通解。在此基础上，给出了当初始盈余等于储备水平或等于零时Gerber-Shiu函数的精确解。这些解是一般情况下Gerber-Shiu函数精确解的关键。作为应用，我们得到了索赔额服从指数分布时零折扣Gerber-Shiu函数的精确解和索赔额服从Erlang（2）分布时破产概率的精确解.最后，我们用数值例子来说明利率和流动准备金对破产概率的影响。
We modify the compound Poisson surplus model for an insurer by including liquid reserves and interest on the surplus. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which do not earn interest. When the surplus attains the level, the excess of the surplus over the level will receive interest at a constant rate. If the level goes to infinity, the modified model is reduced to the classical compound Poisson risk model. If the level is set to zero, the modified model becomes the compound Poisson risk model with interest. We study ruin probability and other quantities related to ruin in the modified compound Poisson surplus model by the Gerber–Shiu function and discuss the impact of interest and liquid reserves on the ruin probability, the deficit at ruin, and other ruin quantities. First, we derive a system of integro-differential equations for the Gerber–Shiu function. By solving the system of equations, we obtain the general solution for the Gerber–Shiu function. Then, we give the exact solutions for the Gerber–Shiu function when the initial surplus is equal to the liquid reserve level or equal to zero. These solutions are the key to the exact solution for the Gerber–Shiu function in general cases. As applications, we derive the exact solution for the zero discounted Gerber–Shiu function when claim sizes are exponentially distributed and the exact solution for the ruin probability when claim sizes have Erlang(2) distributions. Finally, we use numerical examples to illustrate the impact of interest and liquid reserves on the ruin probability.