中文摘要：

本文研究了具有随机保费收入的风险模型的Gerber-Shiu罚金函数的可微性以及渐近性质,随机保费收入通过一个复合泊松过程刻画.本文得到了Gerber-Shiu函数所满足的积分微分方程,给出了Gerber-Shiu罚金函数二次可微与三次可微的充分条件.当所讨论的罚金函数是三次可微的时候,前述积分微分方程可以转化为一般的常微分方程.利用常微分方程的标准方法,当个体随机保费和随机理赔都是指数分布的时候,得到了绝对破产概率在初始盈余趋向于无穷大时的渐近性质.

英文摘要：

In this paper, the differentiability and asymptotic properties of Gerber-Shiu expected discounted penalty function （Gerber-Shiu function for short） associated with the absolute ruin time are investigated, where the risk model is given by classical risk model with additional random premium incomes. The additional random premium income process is specified by a compound Poisson process. A couple of integro-differential equations satisfied by Gerber-Shiu function are derived, several sufficient conditions which guarantee the second-order or third-order differentiability of Gerber-Shiu function are provided. Based on the differentiability results, when the individual claim and premium income are both exponential distribution, the previous integro-differential equations can be deduced into a third-order constant ordinary differential equation （ODE for short）. With the standard techniques on ODE, we find the asymptotic behavior of absolute ruin probability when the initial surplus tends to infinity.