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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.

A risk-neutral measure $Q$ is a probability measure equivalent to the physical probability measure $P$, meaning they agree on which events have zero probability. For assets without intermediate payments, prices expressed in the chosen numéraire must be martingales under $Q$. For a positive bank account $B$ and a stock without dividends, this requires $S/B$ to be a martingale; with dividends, use discounted dividend gains. In an unrestricted finite-state one-period financial market, existence of such a measure is equivalent to absence of arbitrage, by the fundamental theorem of asset pricing.

Physical probability $p=P(\text{up})$ describes the modeled frequency or belief about an up move. The risk-neutral probability $q=Q(\text{up})$ is constrained by traded prices. In the worked binomial market, $q=5/8$ even if $p=1/2$ or $p=4/5$. Calling $Q$ risk-neutral does not assume that investors have no risk aversion; it changes the weights used for risk-neutral pricing.