双Poisson风险模型下Gerber-Shiu函数及测度变换的研究

在轻尾假设下,对保险公司盈余离散模型的期望折现罚金函数进行了研究.通过构造指数鞅,定义了新的测度.利用测度变换公式,消去了折现,得到新的期望折现罚金函数,简化了表达式,并且得到了其满足的更新方程.通过新测度下的期望折现罚金函数,得到Lundberg不等式；并利用测度变换,使得新测度下破产的发生变得确定,更新方程将简化为一般更新方程；进而利用关键更新定理,得到了当初始资本趋于无穷大时,期望折现罚金函数的渐进性；最后对于个体索赔额服从指数分布的特殊情况,导出其破产概率公式的显示表达式.

On the Gerber-Shiu function of double Poisson risk model and change of measure

In this paper, the Gerber-Shiu function for a discrete risk models is considered for the surplus of an insurance company mainly under some light-tail assumptions. By a change of measure we remove the discounting, which simplifies the expression, and this leads to (defective) renewal equation that had been found. By the new Gerber-Shiu function, the Lundberg inequalities can easily be obtained. If we use the change of measure such that ruin becomes certain, the renewal equations simplify to ordinary renewal equations. This helps to discuss the asymptotic as the initial capital goes to infinity. For exponential distribution claim sizes, explicit formula for the ruin probability can be derived.