I'm trying to apply Itô Formula to $F(B_t,S_t) = (S_t-B_t)^2$ where $S_t = \max_{0\leq s\leq t} B_s$, but first I need to know if $S_t$ is a continuous semimartingale. I know that $B_t$ is, since $B_t^2-t$ is a martingale and $B_t = B_t + 0$ being $0$ of bounded variation. However, is the maximum of a continuous semimartingale a semimartingale itself?
Any help, any hint is welcome. Thank you very much!