Let $$M(t) = \int_{0}^t Y (u)dB(u).$$ where $$E \left[ \int_{0}^t Y^2(u)du\right] < \infty.$$ Use Itô’s rule to find the differential $dQ$ of the process $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ and to conclude that the process $Q$ is a martingale. In particular, argue that this implies $$E[M^2(t)] = E\left[ \int_{0}^t Y^2(u)du\right].$$
I just learn stochastic calculus recently, how should I tackle the $d(M^2(t))$, in order to get $dQ$? And what should I do next.
p.s. The proof for martingale part (Done.)
Applying Itô's formula (for $f(x)=x^2$) yields
$$\begin{align*} M_t^2-M_0^2 &= 2 \int_0^t M_s \, dM_s + \int_0^t \langle M \rangle_s \\ &= 2 \int_0^t M_s Y_s \, dB_s + \int_0^t Y_s^2 \, ds \end{align*}$$
where we have used that
$$\langle M \rangle_s = Y_s^2 \, ds.$$
Consequently,
$$Q_t = 2 \int_0^t M_s Y_s \, dB_s,$$
i.e.
$$dQ_t = 2 M_t Y_t \, dB_t.$$