= Bounded diffusion coefficient gives square martingales
{title2=$X_t^2-\int_0^t\sigma(X_s)^2ds$}
For a zero-starting solution with $|\sigma|\le C$, the <Itô isometry> gives $\mathbb EX_t^2\le C^2t$, so $X$ is a true <martingale> on every finite horizon. The <Itô formula> represents the displayed process as $2\int_0^tX_s\sigma(X_s)dB_s$, whose expected squared integrand integral is at most $2C^4t^2$. Thus it too is a true square-integrable <martingale>. No <continuity> or Lipschitz condition on the coefficient is needed once a solution is given.
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