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

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 $0<t\le \tau$.

Suppose an equivalent measure $Q$ made $S$ a local martingale. Stop at the positive time furnished by part (i), chosen so that $S_{t\wedge \tau }\ge 0$. A nonnegative local martingale is a supermartingale, and this one starts from $S_0=0$. Hence $S_{t\wedge \tau }=0$ $Q$-almost surely for every deterministic $t$.

But $Q$ and $P$ have the same null events, while $\tau >0$ almost surely. Some positive rational $q$ therefore satisfies $Q(q<\tau )>0$, and on that event part (i) gives $S_q>0$, a contradiction. No such equivalent measure exists.