Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 6 / a / Solution

Approximate $h$ in $L^2[0,t]$ by deterministic step functions $h_n$. Each integral $\int h_{n}dB$ is a linear combination of independent Gaussian increments and hence is Gaussian, with mean zero and variance $\int h_n^2$. The Itô isometry gives convergence in $L^2$ to $\int hdB$, so characteristic functions pass to the limit. Thus