Invariance times

On a probability space $(Ω,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $θ$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,θ]$. In this setup it is well-known that, for any $\mathbb{F}$ semimartingale $X$, the process $X^{θ-}$ ($X$ stopped "right before $θ$") is a $\mathbb{G}$ semimartingale.Given a positive constant $T$, we call $θ$ an invariance time if there exist...