= Brownian sign transform
{c}
{title2=$\int_0^t\operatorname{sgn}(B_s)\,dB_s$}
The sign transform of standard <Brownian motion> is itself standard Brownian motion. The integrand is <predictable> because it is a Borel function of a continuous adapted process. Brownian motion spends zero Lebesgue time at a specified level: integrate $\mathbb P(B_s=0)=0$ over positive times. Consequently the transform's <quadratic variation> is $t$, regardless of the chosen sign value at zero. The <Lévy characterization of Brownian motion> applies.
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