Invariance times

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