In this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an \(\operatorname{Erlang}(n)\) distribution. If the observed surplus level is greater than zero but less than injection line \(b_{1} > 0\), the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line \(b_{1}\). If the observed surplus level is larger than dividend line \(b_{2}\) (\(b_{2} > b_{1}\)), any excess of the surplus over \(b_{2}\) is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.

在本文中，我们通过复合Poisson风险模型对保险公司的盈余进行建模，该盈余过程只能在随机观察时间进行观察。假设保险公司定期观察其盈余水平，以决定在具有\（\ operatorname {Erlang}（n）\）分布的互观察期间的股利支付和注资。如果观察到的盈余水平大于零但小于注入线\（b_ {1}> 0 \），则股东应立即注入一定数量的资本，以使盈余水平回到注入线\（b_ {1 } \）。如果观察到的盈余水平大于股息线\（b_ {2} \）（\（b_ {2}> b_ {1} \）），则超出\（b_ {2} \）的任何盈余立即作为股息分配给公司的股东。当观察到的盈余水平为负时宣布破产。我们导出Gerber–Shiu函数，预期的注资注资和预期的股利支付折现率的明确表述。还给出了数值说明，以分析随机观察时间对精算量的影响。