Before revision:  Since the death time and the default time  can be seen when they occur only, the progressive enlargement of \mathbb F with \tau , that we denote by \mathbb G, sounds taylor-fit for modelling the new flow of information that incorporates both \mathbb F and \tau.
After revision:  Since the death time and the default time 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.