Let $(X_t)$ be a semimartingale on $[0,T]$. Let $Y$ be a random variable and $f$ a real continuous function.
If $Z_t=X_t + f(t)$, under what condition on $f$ does $[Z]_t$ exist ?
If $Z_t=X_t + Yf(t)$, under what condition on $f$ and $Y$ does $[Z]_t$ exist ?
My guess is that $f$ must be of bounded total variation and no condition on $Y$ is needed. How to show that ? I have a hard time understanding $[f]_t$, is it the total variation of $f$ ?