Discounted dividend gains ($Y_t=S_t/B_t+{\int }_0^t{D}_s/B_{s}ds$)

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 $dX=\varphi dB+\pi \cdot (dS+Ddt)$ and using $X=\varphi B+\pi \cdot S$. 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.

For an integrable discrete trading strategy under an equivalent martingale measure, discounted portfolio wealth plus accumulated discounted strategy payments is a martingale. Conditioning its terminal value gives the displayed identity. A formula involving future payments alone requires terminal liquidation, $V_n=0$, or inclusion of that terminal value in the final payment. For an attainable claim paid only at maturity, liquidation yields its usual conditional discounted expected value.

A share paying a continuous dividend rate ${\delta }_t$ produces dollar gains $d{S}_t+{\delta }_{t}dt$. If its price follows a geometric Brownian motion with drift ${\mu }_{\mathrm{price}}$ and its dividend yield ${\delta }_t/S_t$ is constant, its drift in an investment-consumption problem is their sum. The self-financing portfolio equation uses this total return, even if the agent consumes the dividends rather than reinvesting them. Omitting the dividend yield changes both the optimal stock fraction and the equilibrium interest rate.