  Our financial setting consists of a market model with two flows of
information. The smallest flow F is the "public" flow of information which is
available to all agents, while the larger flow G has additional information
about the occurrence of a random time T. This random time can model the default
time in credit risk or death time in life insurance. Hence the filtration G is
the progressive enlargement of F with T. In this framework, under some mild
assumptions on the pair (F, T), we describe explicitly how G-local martingales
can be represented in terms of F-local martingale and parameters of T. This
representation complements Choulli, Daveloose and Vanmaele
\cite{ChoulliDavelooseVanmaele} to the case when martingales live "after T".
The application of these results to the explicit parametrization of all
deflators under G is fully elaborated. The results are illustrated on the case
of jump-diffusion model and the discrete-time market model.
