This is from the following post : https://almostsuremath.com/2010/05/25/the-martingale-representation-theorem/#comment-12778
Let $B=(B^1, \dots, B^d)$ be a $d$-dimensional Brownian motion defined on the natural filtration $(\mathcal{F}_t)_{t\ge 0})$. Let $\alpha^j$ be deterministic processes satisfying $\int_0^\infty (\alpha_t^j)^2dt<\infty$. $U$ a $\mathcal{F}_0$ measurable random variable. And $i$ is the imaginary number.
Let $$Z = U \exp(i\sum_j \int_0^\infty \alpha^j dB^j)$$ and then we can write $$E[Z|\mathcal{F}_t]=U\exp(i \sum_j \int_0^t \alpha^j dB^j - \frac{1}{2}\sum_j \int_t^\infty (\alpha_s^j)^2 ds).$$
Now if we set $M_t = E[Z|\mathcal{F}_t]$,
how do we use the Ito's lemma here to get $$M_t = M_0 + i\sum_j \int_0^t M\alpha^j dB^j?$$
Moreover, if we set $\xi^j = iM\alpha^j$, then how do we ensure that $\int_0^t (\xi_s^i)^2 ds<\infty$ almost surely for each $t\ge 0$?
I would greatly appreciate if anyone could explain these two questions to me.
