双边跳风险过程Gerber-Shiu函数的重尾扰动位相型逼近
Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps
摘要：
本文考虑了一个风险准备金过程，其中索赔和收益服从两个独立的Poisson过程。虽然增益大小是相位型分布，我们假设索赔额是相位型扰动的重尾分量，也就是说，索赔额分布形式上选择为相位型与大概率$1-ε$和重尾与小概率$ε$。我们分析了种子Gerber-Shiu函数编码的联合分布的时间破产，破产前的盈余，破产时的赤字。我们推导出它的值作为一个扩展相对于权力的$ε$与已知的系数，我们构建近似从上述系列的前两项。其主要思想是基于所谓的流体嵌入，允许将所考虑的风险过程放入谱负马尔可夫-可加过程的框架中，并使用Ivanovs和Palmowski（2012）开发的波动理论。
英文摘要：
We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability $1-ε$ and heavy-tailed with small probability $ε$. We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of $ε$ with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in Ivanovs and Palmowski (2012).
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