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

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.