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 Λ 0 = 0 and N t e − Λ t ≤ 1 , we construct a probability measure Q Z and a random time τ such that Q | F ∞ = P | F ∞ and Q [ τ > t | F t ] = Z t . The probability Q Z is linked with the well-known Cox model by an explicit density function. Various properties exist, which characterize Q Z from others. Let G = ( G t ) t ≥ 0 with G t = F t ∨ σ ( { τ ≤ s } : s ≤ t ) . We establish the ( H ′ ) -property between the filtrations F and G , and we provide the enlargement of filtration formula.