 Solution

Let ${\tau }_n$ localize $M$. The Itô isometry givesThus the stopped variables are bounded in $L^2$, and localization plus weak compactness shows that $M$ is a true martingale. The Itô formula applied to $f(x)=x^2$ givesAfter localization this is a martingale; the Burkholder-Davis-Gundy inequalities and $\mathbb{E}[M{]}_t<\infty$ supply the required local integrability, so it is a true martingale.