= Discounted dividend gains
{title2=$Y_t=S_t/B_t+\int_0^tD_s/B_s\,ds$}
If $B$ is a positive riskless <numéraire> of finite variation, $S$ the ex-dividend price vector and $D$ its cash-dividend rate, a <self-financing portfolio> with risky holdings $\pi$ satisfies $d(X/B)=\pi\cdot dY$. This follows by differentiating $X/B$, substituting $dX=\phi\,dB+\pi\cdot(dS+D\,dt)$ and using $X=\phi B+\pi\cdot S$. If $Y$ is a <martingale> under an equivalent measure, discounted wealth is a <local martingale>. A deterministic lower bound on discounted wealth makes it a <supermartingale>, ruling out <arbitrage>. The cumulative dividends here are discounted at their own payment times.
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