Invariance times

On a probability space (.,A,Q) , we consider two filtrations F.G and a G stopping time . such that the G predictable processes coincide with F predictable processes on (0,.]. In this setup, it is well known that, for any F semimartingale X, the process X.. (X stopped .right before ..) is a G semimartingale. Given a positive constant T, we call . an invariance time if there exists a probability measure P equivalent to Q on FT such that, for any (F,P) local martingale X, X.. is a (G,Q) local martingale. We characterize invariance times in terms of the (F,Q) Azéma supermartingale of . and we derive a mild and tractable invariance time sufficiency condition. We discuss invariance times in mathematical finance and BSDE applications.