We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case, the stopping region that characterizes the optimal stopping time is either a half‐line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided.

中文翻译：

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泊松行使机会下美国期权的双重延续区域

我们考虑永续美国看涨期权的Lévy模型，并在Poisson观察结果中将期权的负折现率设置为负。与连续观察的情况类似，表征最佳停止时间的停止区域为半线或间隔。本文的目的是获得针对停止和连续区域以及值函数的显式表达式，着重于频谱上的正负情况。为此，我们通过尺度函数将与第一个泊松到达时间有关的身份计算到一个区间，然后将这些身份应用于最优策略的计算。当观察率增加到无穷大时，我们还讨论了在连续观察情况下最优解的收敛性。还提供了数值实验。