Two optimal stopping problems for geometric random walks with the observer’s power payoff function, on the finite and infinite horizons, are solved. For these problems, an explicit form of the cut value and also optimal stopping rules are established. It is proved that the optimal stopping rules are nonrandomized thresholds and describe the corresponding free boundary. An explicit form of the free boundary is presented.

中文翻译：

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具有功率释放功能的几何随机游走的最佳停止时间

解决了在有限和无限地平线上具有观察者功率支付函数的几何随机游走的两个最佳停止问题。针对这些问题，建立了明确的切割值形式以及最佳的停止规则。证明了最佳停止规则是非随机阈值，并描述了相应的自由边界。提出了自由边界的明确形式。