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

Given a filtered probability space , an -adapted continuous increasing process [Lambda] and a positive local martingale N such that satisfies Zt<=1,t>=0, we construct probability measures and a random time [tau] on an extension of , such that the survival probability of [tau], i.e., is equal to Zt 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 where is the [sigma]-field completed with the -negligible sets. We show that all martingales remain -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.