Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 3 / b / Solution

The process $W_t=a^{-1}{B}_{a^{2}t}$ is standard Brownian motion, by Brownian scaling. Its first exit from $(-1,1)$ occurs at ${\sigma }_a/a^2$, soThe first part ensures integrability. Set $C=\mathbb{E}{\sigma }_1$; it is finite and strictly positive because path continuity at zero gives ${\sigma }_1>0$ almost surely. ThereforeWe keep $C$ unspecified until the comparison of limits requested below.