Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 3 / b / iii

First suppose $0<r<\rho =|z_0|$. For every $R>\rho$, the planar Brownian annulus hitting probability givesHence ${\tau }_r<\infty$ almost surely. This uses increasing outer radii; letting the inner radius tend to zero instead proves the different claim in part (ii).

For $r>\rho$, the disk-exit estimate in (i), with outer radius $r$, directly yields $\mathbb{E}{\tau }_r\le (r^2-{\rho }^2)/2<\infty$.

For $r=\rho$, retain the definition ${\tau }_r=\inf \{t>0:|B_t|=r\}$. The first case shows that radius $\rho /2$ is reached in finite time. Starting there, the Strong Markov property of Brownian motion and the disk-exit argument show that radius $\rho$ is subsequently reached in finite time. Thus the infimum over positive hitting times is finite in this case as well. Altogether,The conclusion can also be made simultaneous in all $r>0$: intersect the probability-one events for positive rational radii, then use continuity to cross each intermediate radius.