We provide a general model of default time, extending the models of Jiao and Li (modelling sovereign risks) and Gehmlich and Schmidt (dynamic defaultable term structure modelling beyond intensity paradigm).

 We show that any  random time $/tau$  can be decomposed in two parts as $/tau= /tau_1 /wedge /tau_2$ under the condition that the first random time  $/tau_1$ avoids stopping times in the reference filtration $/mathbb F$, and the second time $/tau_2$ is thin, i.e., its   graph   is included in a countable union of graphs of stopping times in the reference filtration $/mathbb F$. Under the condition  $/tau_1/vee /tau_2=/infty$, the decomposition is unique. This decomposition is based on a study of the dual optional projection of $/tau$, as the decomposition of a stopping time into accessible and totally inaccessible is based on the dual predictable projection. We show that for a thin time $/tau_2$, any  $/mathbb F$-martingale is a semimartingale in  its progressive enlargement with $/tau_2$ and we give its semimartingale decomposition.

We prove that  any martingale in the reference filtration is a semimartingale in the progressive enlargement with $/tau$ if and only if the same property holds for the progressive enlargement with $/tau_1$ and we give  its semimartingale representation.

We establish in that  the immersion property holds for $/tau$  if and only if it  holds for $/tau_1$.

(Joint work with Anna Aksamit, Tahir Choulli)

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