Past exam of the Brownian motion in complex analysis course of the University of Oxford / 2011 / 3 / a

A bounded holomorphic function of planar Brownian motion is a local martingale up to exit, because its real and imaginary parts are harmonic and Itô's formula has zero drift. Stop before the given $S<T_D$. Localization inside compact subdisks gives the local martingale $f(Z_{t\wedge S}^z)$, and $|f|<1$ makes it bounded. The bounded local martingale criterion therefore gives a true martingale.

Exit from the unit disk is finite almost surely, so $S<T_D$ is finite too. Taking $t\to \infty$ and using the dominated convergence theorem yieldsThe same proof works for any bounded holomorphic function, after rescaling if needed; it does not require the stopping time itself to be bounded.