With probability one, paths of semimartingales have unbounded variation.
What I know is that a martingale is also a semi-martingale, for example, Brownian motion. Hence, Brownian is an example of semi-martingale with unbounded variation. Is the above statement generally ture? If so, how to show it? Otherwise, what is a counter example? Thank you!
The above statement is not true. Semimartingales are a quite general class of processes containing processes of both bounded and unbounded variation. For example a constant process $(X_t) \equiv 0 $ is a semimartingale.