We investigate the Gerber-Shiu discounted penalty function for Markov-modulated Lévy risk processes with random incomes. Firstly, we consider the case when the downward and upward jumps (respectively, claims and random gains) are given by independent compound Poisson processes, with claim sizes with a general distribution function and gains in such a way that their distribution has a rational Laplace transform. Afterwards, we use the above results and weak convergence techniques to study the case when the claims are given by a subordinator and, subsequently, we establish results when the claims are governed by a pure spectrally positive Lévy jump process. Some numerical examples are presented in order to illustrate our results.

我们研究了带有随机收入的马尔可夫调制的 Lévy 风险过程的 Gerber-Shiu 贴现惩罚函数。首先，我们考虑向下和向上跳跃（分别是索赔和随机收益）由独立的复合泊松过程给出的情况，索赔大小具有一般分布函数和增益，其分布具有合理的拉普拉斯变换. 之后，我们使用上述结果和弱收敛技术来研究索赔由下属给出的情况，随后，我们建立了索赔受纯谱正 Lévy 跳跃过程控制的结果。给出了一些数值例子来说明我们的结果。