Orthogonality of the radial martingale and planar Brownian area
= Orthogonality of the radial martingale and planar Brownian area
{title2=$\langle H,Z\rangle=0,\quad\langle H\rangle=\langle Z\rangle=A$}
The radial martingale $H=\int X\,dX+Y\,dY$ and the <planar Brownian stochastic area> $Z=\int Y\,dX-X\,dY$ have common bracket $A=\int(X^2+Y^2)ds$ and zero cross variation, because their integrand vectors $(X,Y)$ and $(Y,-X)$ are orthogonal. The Itô formula also gives $X_t^2+Y_t^2=2H_t+2t$.