In our homework assignment, we are supposed to prove:
If $ M $ is a countinuous local martingale and if for each $ T > 0, E[\sup_{t \leq T } |M_t|] < + \infty $ and $ H^T $ is a bounded predictable process, then $ H \cdot M $ is a true martingale.
In the hint it says that we should use the previous exercise, which was:
If $ M $ is a countinuous local martingale and $ H $ a predictable process such that
for each $ T > 0 $ we have
$ E[\sqrt{\int_0^T H_s^2 d \langle M \rangle_s }] < + \infty $,
then $ H \cdot M $ is a true martingale.
Well, I started like this:
$ | H^T | \leq C \ \Longrightarrow \ E[\sqrt{\int_0^T H_s^2 d \langle M \rangle_s }] \ \leq C E[\sqrt{ \langle M \rangle_T }] $
Moreover I know that $ E[\sup_{t \leq T } |M_t|] < + \infty, \forall T > 0, $ implies that $ M $ is a true martingale (but I don't know if we need this at all).
Anyway, I don't know how to proceed from here, or if the above was the right way to start at all.
Thanks for all your efforts! Regards, Si