We adapt ideas and concepts developed in optimal transport (and its martingale variant) to give a geometric description of optimal stopping times $$\tau $$τ of Brownian motion subject to the constraint that the distribution of $$\tau $$τ is a given probability $$\mu $$μ. The methods work for a large class of cost processes. (At a minimum we need the cost process to be measurable and $$(\mathcal {F}^0_{t})_{t \ge 0}$$(Ft0)t≥0-adapted. Continuity assumptions can be used to guarantee existence of solutions.) We find that for many of the cost processes one can come up with, the solution is given by the first hitting time of a barrier in a suitable phase space. As a by-product we recover classical solutions of the inverse first passage time problem/Shiryaev’s problem.

中文翻译：

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分布约束最优停止问题的几何

我们采用最优传输（及其 Martingale 变体）中开发的想法和概念来给出布朗运动的最优停止时间 $$\tau $$τ 的几何描述，该约束受 $$\tau $$τ 的分布为一个给定的概率 $$\mu $$μ。这些方法适用于一大类成本流程。（至少我们需要成本过程是可测量的并且 $$(\mathcal {F}^0_{t})_{t \ge 0}$$(Ft0)t≥0-适应。可以使用连续性假设以保证解决方案的存在。）我们发现，对于许多可以提出的成本过程，解决方案是由合适的相空间中的障碍的第一次击中时间给出的。作为副产品，我们恢复了逆第一通道时间问题/Shiryaev 问题的经典解决方案。