  This work is concerned with the theory of initial and progressive
enlargements of a reference filtration F with a random time {\tau}. We provide,
under an equivalence assumption, slightly stronger than the absolute continuity
assumption of Jacod, alternative proofs to results concerning canonical
decomposition of an 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.
