Unit-magnitude scalar diffusion has Brownian law ($d{X}_t={\sigma }_{t}d{B}_t,|{\sigma }_t|=1$)

A progressively measurable real diffusion coefficient of absolute value one gives a continuous local martingale with quadratic variation ${\int }_0^t{\sigma }_s^{2}ds=t$. With zero initial value, Lévy characterization of Brownian motion makes the solution standard Brownian motion. This applies to every weak solution of $d{X}_t=\mathrm{sgn}(X_t-a)d{B}_t$. The Brownian law does not imply that the solution equals the given driver or establish strong existence for a discontinuous coefficient.