  We provide a characterization of the family of non-negative local martingales
that have continuous running supremum and vanish at infinity. This is done by
describing the class of random times that identify the times of maximum of such
processes. In this way we extend to the case of general filtrations a result
proved by Nikeghbali and Yor [NY06] for continuous filtrations. Our
generalization is complementary to the one presented by Kardaras [Kar14], and
is obtained by means of similar tools.
