Invariance times

On a probability space $(\mathrm{\Omega },\mathcal{A},\mathbb{Q})$ 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 $\mathcal{F}\mathrm{\_}T$ 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.