 Solution

Put $M_t={\int }_0^t{H}_{s}d{B}_s$ and $A_t={\int }_0^t{H}_s^{2}ds=[M{]}_t$. The assumed path properties make $A$ continuous, strictly increasing and unbounded. Therefore $T$ is a finite stopping time and $A_T={\sigma }^2$, even though its definition uses a strict inequality.

For $u\in \mathbb{R}$, the Itô formula givesIts real and imaginary parts are local martingales, and $|E_t|\le e^{u^2{\sigma }^2/2}$. Thus $E$ is a bounded true martingale. Continuity of $M$, finiteness of $T$, and dominated convergence yield $\mathbb{E}{E}_T=1$. Rearranging givesThe characteristic function uniquely identifies the distribution, proving the Gaussian terminal value at a deterministic bracket level:This proof does not use the time-change theorem that the later part asks us to prove.