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A general result (Peskir and Shiryaev: Optimal Stopping and Free Boundary Problems, 2006, Thm. 2.4, Page 37) is that the solution to an optimal stopping problem $\sup_\tau EG(X_\tau)$ where $X$ is Markov, is the smallest superharmonic function dominating $G(x)$.

This ... doesn't make sense. Superharmonic functions are concave, yet I know of solutions to optimal stopping problems that most certainly are convex (e.g. infinite horizon american put option)! So how does this characterization make sense?

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