Optimal stopping of stochastic transport minimizing submartingale costs

Ghoussoub, Nassif; Kim, Young-Heon; Palmer, Aaron

doi:10.1090/tran/8458

Optimal stopping of stochastic transport minimizing submartingale costs
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by Nassif Ghoussoub, Young-Heon Kim and Aaron Zeff Palmer PDF

Trans. Amer. Math. Soc. 374 (2021), 6963-6989 Request permission

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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