Let $\Delta_{n}$ be a sequence of unbounded sequences $0=t_{0} \leq ...$ s.t. $\|\Delta_{n}\| \rightarrow 0$. Let $X$ be a continuous semimartingale and $H$ be continuous and X-integrable. Is it true that
$\sum_{k \in \mathbb{N}} H_{t_{k-1}} (X_{t \wedge t_{k}}-X_{t \wedge t_{k-1}})^2 \rightarrow \int H d\langle X\rangle$ in ucp?