= Solution
{c}
The two-dimensional <Itô formula> gives the local martingale part of $u(X_t,Y_t)$ as $\int_0^t u_x\,dX+\int_0^t u_y\,dY$, with all derivatives evaluated along $(X_s,Y_s)$. Independent Brownian coordinates have brackets $[X]_t=[Y]_t=t$ and $[X,Y]_t=0$. Hence the <quadratic variation> is
$$
\boxed{\langle u(X,Y)\rangle_t
=\int_0^t(u_x^2+u_y^2)(X_s,Y_s)\,ds.}
$$
Similarly, the <quadratic covariation> with $v(X,Y)$ is
$$
\langle u(X,Y),v(X,Y)\rangle_t
=\int_0^t(u_xv_x+u_yv_y)(X_s,Y_s)\,ds.
$$
Sufficient pointwise conditions are equal gradient lengths and perpendicular gradients:
$$
u_x^2+u_y^2=v_x^2+v_y^2,\qquad u_xv_x+u_yv_y=0.
$$
The <Cauchy-Riemann equations> $u_x=v_y$, $u_y=-v_x$ imply both. These are the <quadratic covariations of an analytic Brownian image>; harmonicity is needed to remove the drift, but the two bracket identities themselves follow just from the displayed derivative relations.
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