An Explicit Model of Default Time with given Survival Probability∗

For a given filtered probability space (Ω,F,P), where F = (Ft)t≥0 is a filtration, an F-adapted continuous increasing process Λ and a positive P-F local martingale N such that Zt := Nte−Λt ≤ 1, t ≥ 0, we construct a model of default time, i.e., a probability measure QZ and a random time τ on an extension of (Ω,F,P), such that Q[τ > t|Ft] = Zt, t ≥ 0. The probability QZ is linked with the well-known Cox model by an explicite density function. There exist various properties, such as the proportionality or the invariance of conditional laws, which characterize QZ from others. We will make a particular attention on the complete separation property. As usual the model is equipped with an enlarged filtration G = (Gt)t≥0 with Gt = Ft ∨ σ(τ ∧ t). We will show that all P-F martingales are QZ-G semimartingale and give an explicite semimartingale decomposition formula.