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 τ can be decomposed in two parts as τ = τ 1 ∧τ 2
under the condition that the first random time τ 1 avoids stopping times in the ref-
erence filtration F, and the second time τ 2 is thin, i.e., its graph is included in a
countable union of graphs of stopping times in the reference filtration F. Under the
condition τ 1 ∨ τ 2 = ∞, the decomposition is unique. This decomposition is based
on a study of the dual optional projection of τ , as the decomposition of a stopping
time into accessible and totally inaccessible is based on the dual predictable pro-
jection. We show that for a thin time τ 2 , any F-martingale is a semimartingale in
its progressive enlargement with τ 2 and we give its semimartingale decomposition.
We prove that any martingale in the reference filtration is a semimartingale in
the progressive enlargement with τ if and only if the same property holds for the
progressive enlargement with τ 1 and we give its semimartingale representation.
We establish in that the immersion property holds for τ if and only if it holds
for τ 1 .This is a joint work with Anna Aksamit and Tahir Choulli.