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

Given a filtered probability space (Ω,F=(Ft)t≥0,P)(Ω,F=(Ft)t≥0,P), an FF-adapted continuous increasing process ΛΛ and a positive (P,F)(P,F) local martingale NN such that Zt:=Nte−Λt satisfies Zt≤1,t≥0Zt≤1,t≥0, we construct probability measures QQ and a random time ττ on an extension of (Ω,F,P)(Ω,F,P), such that the survival probability of ττ, i.e., Q[τ>t|Ft]Q[τ>t|Ft] is equal to ZtZt for t≥0t≥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=(Gt)t≥0G=(Gt)t≥0 where GtGt is the σσ-field ∩s>t(Fs∨σ(τ∧s))∩s>t(Fs∨σ(τ∧s)) completed with the QQ-negligible sets. We show that all (P,F)(P,F) martingales remain GG-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.