随机保费风险模型下的平均折现罚金函数

本文研究随机保费风险模型下与破产时刻相关的平均折现罚金函数. 与经典的Cram\'{e}r-Lundberg模型相比这里的保费过程不再是时间的线性函数, 而是一个与理赔独立的复合Possion过程. 我们得到了罚金函数所满足的积分方程, 它提供了一种研究破产量的统一方法. 利用该积分方程我们得到了破产时刻, 破产时赤字, 破产前瞬时盈余的Laplace变换; 并在指数分布的特殊情况下求出了他们的显著表达式, 推广了Boikov (2003)的结论.

On the Expected Discounted Penalty Function Associated with the Time of Ruin for a Risk Model with Random Income

This paper studies the expected discounted penalty function associated with the time of ruin for a risk model with stochastic premium. The premium process is no longer a linear function of time in contrast with the classical Cram\'{e}r-Lundberg model. The aggregate premiums constitute a compound Poisson process which is also independent of the claim process. Integral equation for the penalty function is derived, which provides a unified treatment to the ruin quantities. Applications of the integral equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. In some special cases with exponential distributions, closed form expressions for these quantities are obtained, which generalize some known results about the problems of ruin in Boikov (2003).