  We build a general model for pricing defaultable claims. In addition to the
usual absence of arbitrage assumption, we assume that one defaultable asset (at
least) looses value when the default occurs. We prove that under this
assumption, in some standard market filtrations, default times are totally
inaccessible stopping times; we therefore proceed to a systematic construction
of default times with particular emphasis on totally inaccessible stopping
times. Surprisingly, this abstract mathematical construction, reveals a very
specific and useful way in which default models can be built, using both market
factors and idiosyncratic factors. We then provide all the relevant
characteristics of a default time (i.e. the Az\'ema supermartingale and its
Doob-Meyer decomposition) given the information about these factors. We also
provide explicit formulas for the prices of defaultable claims and analyze the
risk premiums that form in the market in anticipation of losses which occur at
the default event. The usual reduced-form framework is extended in order to
include possible economic shocks, in particular jumps of the recovery process
at the default time. This formulas are not classic and we point out that the
knowledge of the default compensator or the intensity process is not anymore a
sufficient quantity for finding explicit prices, but we need indeed the Az\'ema
supermartingale and its Doob-Meyer decomposition.
