Bounded diffusion coefficient gives square martingales ($X_t^2-{\int }_0^t\sigma (X_s{)}^{2}ds$)

For a zero-starting solution with $|\sigma |\le C$, the Itô isometry gives $\mathbb{E}{X}_t^2\le C^{2}t$, so $X$ is a true martingale on every finite horizon. The Itô formula represents the displayed process as $2{\int }_0^t{X}_s\sigma (X_s)d{B}_s$, whose expected squared integrand integral is at most $2C^4{t}^2$. Thus it too is a true square-integrable martingale. No continuity or Lipschitz condition on the coefficient is needed once a solution is given.