stopping time bounding the process

Stream: Brownian motion

[Etienne Marion]

What are the right hypotheses implying that $\inf \{t | |X_t| \ge n\}$ is a stopping time? The proof of https://almostsuremath.com/2009/12/23/localization/#scn_local_lem9 (which is part of the blueprint) uses this, but it looks like it requires right-continuity of the filtration when $X$ is right continuous, while the blueprint does not mention this.

[Rémy Degenne]

That source assumes right continuous filtrations everywhere. The same applies to our work: assume right continuous filtrations whenever needed (and pretty much any interesting result will need it).

[Etienne Marion]

Noted, thanks!

[Rémy Degenne]

This blueprint result is relevant: https://remydegenne.github.io/brownian-motion/blueprint/chap-debut_theorem.html#lem:isStoppingTime_leastGE

[Etienne Marion]

Yes I had seen that but I was wondering whether the right-continuity of the filtration was necessary in the case where the set is open or closed.