  In this paper, a study of random times on filtered probability spaces is
undertaken. The main message is that, as long as distributional properties of
optional processes up to the random time are involved, there is no loss of
generality in assuming that the random time is actually a randomised stopping
time. This perspective has advantages in both the theoretical and practical
study of optional processes up to random times. Applications are given to
financial mathematics, as well as to the study of the stochastic behaviour of
Brownian motion with drift up to its time of overall maximum as well as up to
last-passage times over finite intervals. Furthermore, a novel proof of the
Jeulin-Yor decomposition formula via Girsanov's theorem is provided.
