= Solution
{c}
Write $Z_t=X_t+iY_t$ and $u=\operatorname{Re}g$. Since $g$ is entire, $u$ is a <harmonic function>, so $u_{xx}+u_{yy}=0$. The two-dimensional <Itô formula> gives
$$
du(Z_t)=u_x(Z_t)\,dX_t+u_y(Z_t)\,dY_t
+\tfrac12(u_{xx}+u_{yy})(Z_t)\,dt.
$$
The drift vanishes. Stop on successively larger disks to bound the derivatives and define the stochastic integrals without global growth assumptions. Thus
$$
\boxed{\operatorname{Re}g(Z_t)\text{ is a local martingale}.}
$$
This is the <harmonic functions of Brownian motion> principle.
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