Abstract:Given a stochastic state process $(X_t)_t$ and a real-valued submartingale cost process $(S_t)_t$, we characterize optimal stopping times $\tau$ that minimize the expectation of $S_\tau$ while realizing given initial and target distributions $\mu$ and $\nu$, i.e., $X_0\sim \mu$ and $X_\tau \sim \nu$. A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair $(X_t, S_t)_t$, which generalizes the twist condition on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in Ghoussoub, Kim, and Palmer [Calc. Var. Partial Differential Equations 58 (2019), Paper No. 113, 31] and Ghoussoub, Kim, and Palmer [A solution to the Monge transport problem for Brownian martingales, 2019] and deals with more general costs.

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中文翻译：

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随机传输的最优停止最小化 submartingale 成本

摘要：给定一个随机状态过程 $(X_t)_t$ 和一个实值 submartingale 成本过程 $(S_t)_t$，我们描述了最优停止时间 $\tau$，在实现给定的同时最小化 $S_\tau$ 的期望初始和目标分布 $\mu$ 和 $\nu$，即 $X_0\sim \mu$ 和 $X_\tau \sim \nu$。考虑并证明在合适的条件下可以实现对偶优化问题。然后，对偶问题的最优解提供了一个接触集，它表征了可以发生最优停止的位置。如果我们假设对 $(X_t, S_t)_t$ 的自然结构假设，则最佳停止时间被唯一确定为该接触集的第一次击中时间，该假设概括了扭曲条件最优运输理论中的成本。本文扩展了 Ghoussoub、Kim 和 Palmer [Calc. 无功 Partial Differential equations 58 (2019), Paper No. 113, 31] 和 Ghoussoub、Kim 和 Palmer [A solution to the Monge transport problem for Brownian martingales, 2019] 并处理更一般的成本。

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