  In this article we study and classify optimal martingales in the dual
formulation of optimal stopping problems. In this respect we distinguish
between weakly optimal and surely optimal martingales. It is shown that the
family of weakly optimal and surely optimal martingales may be quite large. On
the other hand it is shown that the Doob-martingale, that is, the martingale
part of the Snell envelope, is in a certain sense the most robust surely
optimal martingale under random perturbations. This new insight leads to a
novel randomized dual martingale minimization algorithm that doesn't require
nested simulation. As a main feature, in a possibly large family of optimal
martingales the algorithm efficiently selects a martingale that is as close as
possible to the Doob martingale. As a result, one obtains the dual upper bound
for the optimal stopping problem with low variance.
