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.
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