Abstract We present analytic solutions to some optimal stopping problems for the running minimum of a geometric Brownian motion with exponential positive discounting rates. The proof is based on the reduction of the original problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual dual American lookback options with fixed and floating strikes in the Black–Merton–Scholes model from the point of view of short sellers.

中文翻译：

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以正折现率运行最小值的最优停止问题

摘要 我们针对具有指数正折现率的几何布朗运动的运行最小值提出了一些最优停止问题的解析解。证明是基于将原始问题简化为相关的自由边界问题，并通过平滑拟合和法向反射条件解决后者的问题。我们表明，最优停止边界被确定为某些一阶非线性常微分方程的最小解。从卖空者的角度，获得的结果与 Black-Merton-Scholes 模型中具有固定和浮动行使价的永久双重美式回顾期权的估值有关。