For proofing that a càglàd (LCRL) adapted process $X$ is locally bounded, it is usually done by using the stopping time $T_n:=\inf\{t\geq 0: |X_t|>n\} $. For càdlàg (RCLL) adapted processes this doesn't work (as the statement is not true in general), since we don't know anything about the size of the jumps at this point. But if we know that the jumps are bounded, this argument will work - is this correct?
I already often read that the local martingale part of a semimartingale can be taken to be locally bounded, since the jumps larger than $1$ can be 'put into the FV part'. This is what I thought, might be the reason for this.
But if we have a special semimartingale $X=X_0+M+A$ with $M$ local martingale and $A$ predictable FV processes, the processs $M$ can not be taken to be locally bounded in general - is this correct? I suppose this won't work, as the jumps larger than $1$ are not predictable in general. But I'm not sure about this.