Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 3 / d / i / Solution

The given limit inferior says that almost surely there is a random $\delta >0$ such that $t^{-a}{W}_t>-1/2$ whenever $0<t\le \delta$. Thereforeon that interval. The first entrance timeis a stopping time by continuity and satisfies $\tau >0$ almost surely. Before $\tau$ one has $S_t>t^a/2$, and continuity gives $S_{\tau }\ge {\tau }^a/2>0$. Hence $S_t>0$ for every $0<t\le \tau$.