  We present a general model for default time, making precise the role of the
intensity process, and showing that this process allows for a knowledge of the
conditional distribution of the default only "before the default". This lack of
information is crucial while working in a multi-default setting. In a single
default case, the knowledge of the intensity process does not allow to compute
the price of defaultable claims, except in the case where immersion property is
satisfied. We propose in this paper the density approach for default time. The
density process will give a full characterization of the links between the
default time and the reference filtration, in particular "after the default
time". We also investigate the description of martingales in the full
filtration in terms of martingales in the reference filtration, and the impact
of Girsanov transformation on the density and intensity processes, and also on
the immersion property.
