Solution
= Solution
With continuous dividend yield $D$, it is the discounted total gain from the stock, including dividends, that must be a <martingale>. The ex-dividend stock drift under the <risk-neutral measure> is therefore $r-D$. Replace $r-\sigma^2/2$ in the exponential representation by $r-D-\sigma^2/2$ and repeat the chain rule. The discount rate remains $r$, giving
$$
\boxed{V_t+\tfrac12\sigma^2S^2V_{SS}+(r-D)SV_S-rV=0.}
$$
The terminal payoff condition is unchanged.