Here on wikipedia is claimed that the process $X_t:=\sup_{s \in (0,t)} B_s-B_t$$X_t:=\sup_{s \in [0,t]} B_s-B_t$ is distributed like $\vert B_t \rvert$ where $B_t$ is standard Brownian motion.
On the other hand, it is claimed here in Corollary $6.21$ that $\sup_{s \in (0,t)} B_s$$\sup_{s \in [0,t]} B_s$ is distributed like $\vert B_t \rvert.$
So how is it possible that $\sup_{s \in (0,t)} B_s-B_t$$\sup_{s \in [0,t]} B_s-B_t$ is distributed like $\sup_{s \in (0,t)} B_s.$$\sup_{s \in [0,t]} B_s.$ There seems to be something wrong with probability.
If you have any further questions, please let me know.
