Let $B$ be a standard Brownian motion, and,
for some function $f$.
What are the condition on $f$ for $X_t$ to be of finite variation?
Let $Y_t=\int_0^t f(B_s)ds$, if $f$ is continuous then $Y_t$ is of finite variation.
Does it then imply that $X_t$ is of finite variation?
If $f$ is only bounded, I think nothing can be infered.
This question is related.