Before revision:  In this setting, our principal goal resides in describing as explicit as possible the set of all deflators, which constitutes the dual set of all "admissible" wealth processes, for the stopped model S^{\tau death time and the default time (that \tau might represent) can be seen when they occur only, the progressive enlargement of \mathbb F with \tau sounds tailor-fit for modelling the new flow of information  that incorporates both \mathbb F and \tau.
After revision:  In this setting, our principal goal resides in describing as explicit as possible the set of all deflators, which constitutes the dual set of all "admissible" wealth processes, for the stopped model S^{\tau death time and the default time (that \tau might represent) can be seen when they occur only, the progressive enlargement of \mathbb F with \tau sounds tailor-fit for modelling the new flow of information \mathbb G that incorporates both \mathbb F and \tau.