We study the problem of stopping a Brownian bridgeXin order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

中文翻译：

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布朗桥指数的最优停止

我们研究停止布朗桥的问题X为了最大化指数增益函数的期望值。这个问题是由 Ernst 和 Shepp (2015) 提出的，其动机是非负价格的债券出售。由于指数增益的非线性结构，我们不能依靠文献中使用的方法来找到封闭形式解决涉及布朗桥的其他问题。相反，我们必须直接处理时间不均匀扩散的停止问题。我们开发了基于布朗桥的路径特性和最优停止理论的鞅方法的技术，这使我们能够找到最优停止规则并显示价值函数的规律性。