 Black-Scholes equation with claim dividends

If the claim holder receives a cash-flow rate $k(S_t,t)$ before maturity, the self-financing hedge matches the claim's cum-dividend gain $dp+kdt$. Its stock holding is the option delta $p_x$ and its bank-account value is $p-x{p}_x$. Comparing Itô formula with portfolio gains gives the displayed equation, with terminal value equal to the terminal payoff. The positive source term $k$ represents dividends paid by the claim; dividends on the underlying stock would instead alter the stock's pricing drift. Risk-neutral pricing includes both the discounted terminal payoff and the integral of discounted interim payments.