Given $M_t = \int_0^t uB_u dB_u$. For $0\leq s \leq t$, I would like to compute $E[M^2_t | F_s]$.
The closest thing that I can think of is Ito's isometry. But here we have the conditional expectation, which makes it inapplicable. Additionally, while $M_t$ itself is a martingale, $M_t^2$ isn't in general. I would appreciate suggestions how on to solve this problem.