  Given a stochastic structure with a filtration $\mathbb{F}$, the class of all
random times whose conditional distribution functions are differentiable with
respect to some $\mathbb{F}$ adapted non decreasing processes is considered.
The main property of a random time in this class is that it can be
isomorphically implanted into an auxiliary model which is absolutely continuous
with respect to a Cox model. Three formulas are established: the conditional
expectation formula, the optional splitting formula, and the enlargement of
filtration formula. This study is particularly useful for models which are not
defined directly with Cox models, such as the dynamic one-default model
developed recently.
