Optimal stopping of stochastic transport minimizing submartingale costs

Given a stochastic state process $(X_t)_t$ and real-valued submartingale cost $(S_t)_t$, we characterize optimal stopping times $\tau$ that minimize the expectation of $S_\tau$ while realizing given initial target distributions $\mu$ $\nu$, i.e., $X_0\sim \mu$ $X_\tau \sim \nu$. A dual optimization problem is considered shown to be attained under suitable conditions. The solution then provides contact set, which characterizes location where can occur. time uniquely determined as first hitting this set provided assume natural structural assumption on pair $(X_t, S_t)_t$, generalizes