Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 4 / c / Solution

Use the discounted wealth $\tilde{X}=X/B$ and the discounted dividend gainsThe integral uses $ds$: the printed $dt$ inside an integral indexed by $s$ is a differential typo. Since $B$ is a positive finite-variation process, the Itô product rule givesThe bank-account terms cancel using $X_t={\varphi }_t{B}_t+{\pi }_t\cdot S_t$.

If $Y$ is a $Q$-martingale, discounted self-financing wealth is a $Q$-local martingale. The admissible trading strategy lower bound makes it a supermartingale, exactly as in the previous no-arbitrage proof. A zero-cost portfolio with nonnegative terminal wealth must therefore have terminal wealth zero $Q$-almost surely and, by equivalence, physically almost surely. Hence an equivalent martingale measure for discounted dividend gains excludes arbitrage. It is the gains process, not just the ex-dividend price ratio, that must have zero martingale drift.