  In the theory of progressive enlargements of filtrations, the supermartingale
$Z_{t}=\mathbf{P}(g>t\mid \mathcal{F}_{t}) $ associated with an honest time g,
and its additive (Doob-Meyer) decomposition, play an essential role. In this
paper, we propose an alternative approach, using a multiplicative
representation for the supermartingale Z_{t}, based on Doob's maximal identity.
We thus give new examples of progressive enlargements. Moreover, we give, in
our setting, a proof of the decomposition formula for martingales, using
initial enlargement techniques, and use it to obtain some path decompositions
given the maximum or minimum of some processes.
