  On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two
filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time
$\theta$ such that the $\mathbb{G}$ predictable processes coincide with
$\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup it is
well-known that, for any $\mathbb{F}$ semimartingale $X$, the process
$X^{\theta-}$ ($X$ stopped "right before $\theta$") is a $\mathbb{G}$
semimartingale.Given a positive constant $T$, we call $\theta$ an invariance
time if there exists a probability measure $\mathbb{P}$ equivalent to
$\mathbb{Q}$ on $\mathcal{F}\_T$ such that, for any $(\mathbb{F},\mathbb{P})$
local martingale $X$, $X^{\theta-}$ is a $(\mathbb{G},\mathbb{Q})$ local
martingale. We characterize invariance times in terms of the
$(\mathbb{F},\mathbb{Q})$ Az\'ema supermartingale of $\theta$ and we derive a
mild and tractable invariance time sufficiency condition. We discuss invariance
times in mathematical finance and BSDE applications.
