This work is concerned with the theory of initial and progressive enlargements of a reference filtration 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 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.Keywords: initial and progressive enlargements of filtrations, predictable projection, canonical decomposition of semimartingales, predictable representation theorem.

No abstract

* This research benefited from the support of the Europlace Institute of Finance and of the exchange grant of AMaMeF. It was motivated by a presentation of J.-P. Lardy at the CRIS research working group [18] (see http://www.criscreditrisk.com). The authors thank J.-P. Lardy, F. Patras, S. Assefa and other members from the CRIS research group, as well as T. Bielecki, M. Rutkowski and V. Brunel, for enlightening discussions, comments and remarks. † See http://www.cris-creditrisk.com.

No abstract