This work is concerned with the theory of initial and progressive enlargements of a
reference filtration
\hbox{$\mathbb{F}$}
with a random time τ. We provide, under an
equivalence assumption, slightly stronger than the absolute continuity assumption of
Jacod, alternative proofs to results concerning canonical decomposition of an \hbox{$\mathbb{F}$}
-martingale
in the enlarged filtrations. Also, we address martingales’ characterization in the
enlarged filtrations in terms of martingales in the reference filtration, as well as
predictable representation theorems in the enlarged filtrations.