Random time with differentiable conditional distribution function

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.