  Given a random time, we characterize the set of martingales for which the
stopping theorems still hold. We also investigate how the stopping theorems are
modified when we consider arbitrary random times. To this end, we introduce
some families of martingales with remarkable properties. We also investigate,
in the Brownian setting, the relationships between a given random time and the
underlying Brownian Motion in the progressively enlarged filtration with
respect to this random time.
