I have a process $f_t$ adapted to a Weiner process, $W_t$. Moreover, $f_t$ is uniformly bounded by a constant, i.e., $|f_t| \leq C$. How can I bound
$$ \mathbb E \left[ \exp \left( \sup_{0 < u < t} \left | \int_0^u f_s \ d W_s \right| \right) \right] $$
I thought that I could use
$$ \left |\int_0^t f_s \ d W_s \right| \leq C \left|\int_0^t \ d |W|_s \right| = C \ |W_t | $$
However, I cannot quite prove the inequality. In fact, I am not sure it's true.