We consider de Finetti's problem for spectrally one-sided L\'evy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

翻译：我们考虑的是德费内蒂的光谱单向L\'evy风险模型问题,该模型的控制战略在Lebesgue措施方面是绝对持续的。此外,我们考虑的是这一版本,对破坏时间有限制。为了确定上述模式的解决办法,我们首先用报废时的终极价值解决最佳红利问题,并展示临界值战略的最佳性。接下来,我们提出拉格朗加双面问题,并表明补充性松懈条件已经得到满足,体现了最佳拉格朗格乘数的特性。最后,我们用一系列数字例子来说明我们的结论。