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.