  This paper proves the existence of optimal stopping times via elementary
functional analytic arguments. The problem is first relaxed into a convex
optimization problem over a closed convex subset of the unit ball of the dual
of a Banach space. The existence of optimal solutions then follows from the
Banach--Alaoglu compactness theorem and the Krein--Millman theorem on extreme
points of convex sets. This approach seems to give the most general existence
results known to date. Applying convex duality to the relaxed problem gives a
dual problem and optimality conditions in terms of martingales that dominate
the reward process.
