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I have a Brownian motion $\{B(t) : t \in [0,1] \}$ and I'm trying to figure out if

$$ \sup \limits_{t \in [0,1]} \vert B(t) \vert < \infty $$

almost surely holds. Does this immedeately follow as $[0,1]$ is a compact interval and the paths $t \mapsto B(t)$ are almost surely continous, such that the extreme value theorem is applicable?

Thanks!

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Surely. For any $\omega$ the supremum is finite because continuous functions on a compact interval are bounded

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  • $\begingroup$ Thank you Kavi! $\endgroup$ – Bazzan Sep 27 at 9:42

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