Given a process $\sigma_s$ which is cadlag and adapted I need to prove the following identity in law
$$ \int_a^b\sigma_s\,dW_s \stackrel{\mathcal{L}}{=}U\cdot\left(\int_a^b\sigma_s^2\,ds\right)^{1/2},\quad W=\text{ Brownian motion} $$
with $U\sim\text{N}(0,1)$. I think that this should be almost obvious given the definition of stochastic integration, but now I cannot see it anymore.
A possible solution could be: (I am attaching screenshot of my own pdf)
