Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2009 / B / 6 / c / i / Solution

Write $Z_t=X_t+i{Y}_t$ and $u=\mathrm{Re}g$. Since $g$ is entire, $u$ is a harmonic function, so $u_{xx}+u_{yy}=0$. The two-dimensional Itô formula givesThe drift vanishes. Stop on successively larger disks to bound the derivatives and define the stochastic integrals without global growth assumptions. ThusThis is the harmonic functions of Brownian motion principle.