INVARIANCE TIMES

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