$(\mathcal{F_n})_n$ is a filtration. We want to characterize all the applications $\phi:\overline{\mathbb{N}} \to \overline{\mathbb{N}}$ such that for every $\mathcal{F}_n$-stopping time $\theta, \phi(\theta)$ is also a $\mathcal{F}_n$-stopping time.
Let's consider the following property $(P)$: $\exists k \in \overline{\mathbb{N}^*},\phi(n) \geq n$ if $n \leq k$ and $\phi(n)=k$ if $n>k.$
It seems $(P)$ is a necessary and sufficient condition so that for any stopping time $\theta,$ $\phi(\theta)$ is also a stopping time.
More generally for every stopping time $\theta_1,...,\theta_d,$ $\phi(\theta_1,...,\theta_d)$ is a stopping time if and only if $(P)$ is verified for each variable separately (particular cases where $(P)$ is verified for each variable $\theta_1+\theta_2,\max(\theta_1,\theta_2),\min(\theta_1,\theta_2)$).
Do you have a proof for this result? Especially when it comes to prove $(P)$ after supposing that $\phi(\theta)$ is a stopping time