 Solution

A self-financing replicating strategy must fund both the claim's price change and its outgoing cash flow:Apply Itô's formula to $V(t,S_t)$. Matching the diffusion gives $\Delta =V_s$. Matching drift and eliminating $\beta B=V-s{V}_s$ gives the Black-Scholes equation with claim dividendsThe physical drift cancels because stock risk is hedged. The cash-flow term has a plus sign because it is a payment to the holder.