# How can the solution to a optimal stopping problem be superharmonic?

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?