 Solution

For claim dividends proportional to the underlying stock, put $V(S,t)=S\varphi (t)$. Then $V_S=\varphi$, $V_{SS}=0$ and $V_t=S{\varphi }^{\prime }$, so the partial differential equation reduces to the linear ordinary differential equationThe integrating factor $e^{-Dt}$ gives $(e^{-Dt}\varphi {)}^{\prime }=-e^{-Dt}f(t)$. Integrating backwards from the terminal value,The formula assumes the displayed integral is finite. Under the risk-neutral measure, ${\mathbb{E}}_t[S_u]=S{e}^{(r-D)(u-t)}$. Discounting each cash payment at its own payment time gives the same result:The interest rate disappears because every payment is proportional to the stock price.