= Quadratic covariations of an analytic Brownian image
{title2=$[U]_t=[V]_t=\int_0^t|f'(Z_s)|^2ds,\quad[U,V]_t=0$}
For an <analytic function> $f=u+iv$ and planar <Brownian motion> $Z=X+iY$ with <independent> standard coordinates, the <Cauchy-Riemann equations> make $u,v$ harmonic. The <Itô formula> gives $dU=u_xdX+u_ydY$ and $dV=-u_ydX+u_xdY$. Localizing the derivatives proves both coordinates are <local martingales>. Squaring their diffusion coefficients and multiplying the two rows gives the displayed <quadratic variations> and zero <quadratic covariation>.
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