Random times with given survival probability and their -martingale decomposition formula

Given a filtered probability space ( Ω , F = ( F t ) t ≥ 0 , P ) , an F -adapted continuous increasing process Λ and a positive ( P , F ) local martingale N such that Z t : = N t e − Λ t satisfies Z t ≤ 1 , t ≥ 0 , we construct probability measures Q and a random time τ on an extension of ( Ω , F , P ) , such that the survival probability of τ , i.e., Q [ τ > t | F t ] is equal to Z t for t ≥ 0 . We show that there exist several solutions and that an increasing family of martingales, combined with a stochastic differential equation, constitutes a natural way to construct these solutions. Our extended space will be equipped with the enlarged filtration G = ( G t ) t ≥ 0 where G t is the σ -field ∩ s > t ( F s ∨ σ ( τ ∧ s ) ) completed with the Q -negligible sets. We show that all ( P , F ) martingales remain G -semimartingales and we give an explicit semimartingale decomposition formula. Finally, we show how this decomposition formula is intimately linked with the stochastic differential equation introduced before.