The recent papers Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017) and Avram et al. (Mathematics 9(9):931, 2021) cost induced dichotomy for optimal dividends in the cramr-lundberg model. Avram et al. (Mathematics 9(9):931, 2021) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram et al. cost induced dichotomy for optimal dividends in the Cramér-Lundberg model (Avram et al. in Mathematics 9(9):931, 2021). Secondly, it extends the results to the case when a final penalty P is taken into consideration as well besides a proportional cost \(k>1\) for capital injections. This requires amending the “scale and Gerber-Shiu functions” already introduced in Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017). Thirdly, in the exponential case, the results will be made even more explicit by employing the Lambert-W function. This tool has particular importance in computational aspects and can be employed in theoretical aspects such as asymptotics.

最近的论文 Gajek 和 Kucinsky (Insur Math Econ 73:1–19, 2017) 和 Avram 等人。(Mathematics 9(9):931, 2021) cramr-lundberg 模型中最优股息的成本诱导二分法。阿夫拉姆等人。(Mathematics 9(9):931, 2021) 研究了破产时停止注资时优化分红的控制问题。第一篇论文在光谱负 Lévy 模型下工作；第二个在 Cramér-Lundberg 模型下工作，具有指数跳跃，结果更加明确。目前的论文有三个目的。首先，它说明了一个事实，即可以使用 Avram 等人研究的特定指数案例获得一般问题的相当合理的近似值。Cramér-Lundberg 模型中最佳股息的成本诱导二分法（Avram 等人在数学 9(9):931, 2021 中）。除了资本注入的比例成本\(k>1\)外，还考虑了P。这需要修改 Gajek 和 Kucinsky 中已经引入的“比例和 Gerber-Shiu 函数”（Insur Math Econ 73:1-19, 2017）。第三，在指数情况下，通过使用 Lambert-W 函数，结果将更加明确。该工具在计算方面具有特别重要的意义，可用于渐近等理论方面。