Projections of martingales in enlargements of Brownian filtrations under Jacod’s equivalence hypothesis

We consider the initial and progressive enlargements of a Brownian filtration with a random time, that is, a strictly positive random variable. We assume Jacod’s equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the Brownian filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged Brownian filtration, we derive explicit expressions for the components which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the Brownian filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the Brownian filtration.