An explicit model of default time with given survival probability

For a given filtered probability space (Ω,F,P), an F-adapted continuous increasing process Λ and a positive P-F local martingale N such that Λ0=0 and Nte−Λt≤1, we construct a probability measure QZ and a random time τ such that Q|F∞=P|F∞ and Q[τ>t|Ft]=Zt. The probability QZ is linked with the well-known Cox model by an explicit density function. Various properties exist, which characterize QZ from others. Let G=(Gt)t≥0 with Gt=Ft∨σ({τ≤s}:s≤t). We establish the (H′)-property between the filtrations F and G, and we provide the enlargement of filtration formula.