Consider an Ito process $$ X(t) = \int_0^t \tilde B(s)\, dB(s), \quad t\in[0,T], $$ where $B(s),\tilde B(s)$ are independent Brownian motions.
Let $\pi^l=\{0=t_0^l<t_1^l<\dots < t^l_{m(l)}=T\}$ be a sequence of partitions of $[0,T]$ s.t. $\max\limits_{i}(t^l_{i+1}-t^l_i)\to 0$ as $l\to\infty$. I want to prove that
$$
\sup\limits_{l}\sum_{i=0}^{m(l)-1}\big|X(t^l_{i+1}) -X(t^l_{i}) - \tilde B(t^l_i)\big(B(t^l_{i+1}) - B(t^l_{i})\big) \big| < \infty \quad \text{a.s.}
$$
First I thought about estimating
$|\int_{t_i}^{t_{i+1}}\tilde B(s) - \tilde B(t_i)\, dB(s)|$, but haven not found useful majorants.