Planar Brownian stochastic area ($Z_t={\int }_0^t{Y}_{s}d{X}_s-X_{s}d{Y}_s$)

For independent coordinate Brownian motions, the stochastic area with the stated orientation is $Z_t={\int }_0^t{Y}_{s}d{X}_s-X_{s}d{Y}_s$. Its bracket is ${\int }_0^t(X_s^2+Y_s^2)ds$. Reversing orientation changes its sign but not its law or quadratic variation.

The radial martingale $H=\int XdX+YdY$ and the planar Brownian stochastic area $Z=\int YdX-XdY$ 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 $X_t^2+Y_t^2=2H_t+2t$.

Jointly time-change the radial and area martingales by the inverse of $A=\int R^{2}ds$. The bracket matrix becomes $u{I}_2$, so the Lévy characterization of multidimensional Brownian motion gives two independent Brownian coordinates. The clock is strictly increasing; if it were finite at infinity, the radial martingale would converge and $R_t^2=2t+O(1)$ would force its integral to diverge. Thus $R_t^2=2W_{A(t)}+2t$ and $Z_t=B_{A(t)}$ with $W,B$ independent.