Martingale representation on enlarged filtrations: the role of the accessible jump times

We consider a filtration (Formula presented.) obtained as enlargement of a filtration (Formula presented.) by a filtration (Formula presented.). We assume that all (Formula presented.) -local martingales are represented by a martingale M and all (Formula presented.) -local martingales are represented by a martingale N. M and N are not necessarily quasi-left continuous processes and their jump times may overlap. We first analyse the contribution of the accessible jump times of M and N to the Jacod dimension of the space of the (Formula presented.) -martingales. Then we prove a martingale representation theorem on (Formula presented.).