= Solution
{c}
A self-financing replicating strategy must fund both the claim's price change and its outgoing cash flow:
$$
dV_t+q_tS_t\,dt=\Delta_t\,dS_t+\beta_t\,dB_t,\qquad
V_t=\Delta_tS_t+\beta_tB_t.
$$
Apply <Itô's formula> to $V(t,S_t)$. Matching the diffusion gives $\Delta=V_s$. Matching drift and eliminating $\beta B=V-sV_s$ gives the <Black-Scholes equation with claim dividends>
$$
\boxed{V_t+\tfrac12\sigma^2s^2V_{ss}+rsV_s-rV+q_ts=0,\qquad
V(T,s)=\Psi(s).}
$$
The physical drift cancels because stock risk is hedged. The cash-flow term has a plus sign because it is a payment to the holder.
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