Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 2 / c / iii / Solution

Part (ii) represents the process as an Iterated Itô integral:For a centered normal random variable $B_s$ of variance $s$,Consequently, for every finite $T$,The Itô integral of this square-integrable stochastic integrand is a true square-integrable martingale on $[0,T]$. Since $T$ is arbitrary,The last identity is the Itô isometry.