Projections in enlargements of filtrations under Jacod's equivalence hypothesis for marked point processes *.

Summary We consider the initial and progressive enlargements of a filtration generated by a marked point process (called the reference filtration) with a strictly positive random time. We assume Jacod's equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the reference filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged reference filtration, we derive explicit expressions for the coefficients which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the reference filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the reference filtration.