= Unit-magnitude scalar diffusion has Brownian law
{c}
{title2=$dX_t=\sigma_t\,dB_t,\quad |\sigma_t|=1$}
A <progressively measurable> real diffusion coefficient of absolute value one gives a <continuous local martingale> with <quadratic variation> $\int_0^t\sigma_s^2ds=t$. With zero initial value, <Lévy characterization of Brownian motion> makes the solution standard <Brownian motion>. This applies to every <weak solution> of $dX_t=\operatorname{sgn}(X_t-a)dB_t$. The Brownian law does not imply that the solution equals the given driver or establish strong existence for a discontinuous coefficient.
Back to article page
