= Solution
Take a non-dividend-paying <stock> with physical dynamics $dS_t=\mu S_tdt+\sigma S_tdW_t$ and <continuous-time bank account> $B_t=e^{\rho t}$. Write $T=t_0$ and assume the candidate claim price $p(x,t)$ is $C^{2,1}$ before maturity, with enough growth control for the hedge to be admissible. The terminal condition is $p(x,T)=f(x)$.
The <Itô formula> gives the claim's price change
$$
dp(S_t,t)=\left(p_t+\mu S_tp_x+\tfrac12\sigma^2S_t^2p_{xx}\right)dt+\sigma S_tp_x\,dW_t.
$$
A <self-financing portfolio> worth $p$ with $\Delta$ shares has bank-account value $p-\Delta S_t$. Its gain is
$$
\Delta\,dS_t+\rho(p-\Delta S_t)dt
=\{\mu\Delta S_t+\rho(p-\Delta S_t)\}dt+\sigma\Delta S_t\,dW_t.
$$
To replicate the claim, the Brownian coefficients must agree, forcing the <delta hedge> $\Delta=p_x$. Equating the remaining <drift> coefficients cancels the physical <drift> $\mu$ and yields
$$
\boxed{p_t+\rho xp_x+\tfrac12\sigma^2x^2p_{xx}-\rho p=0,\qquad p(x,T)=f(x).}
$$
The cancellation explains why the <Black-Scholes equation> contains the <interest rate>, not the physical <stock> <drift>. Conversely, if an admissible solution of this equation is given, holdings $p_x$ in the <stock> and $(p-xp_x)/B_t$ in the <bank account> have value $p$ and gains exactly $dp$. They are therefore a <self-financing portfolio> replicating the terminal payoff. Absence of <arbitrage> forces the claim to have this price. Merely decomposing the current value into arbitrary holdings would not justify self-financing; the gain identity and <delta hedge> are essential.
Now suppose the claim holder receives the cash rate $k(S_t,t)$ before $T$. The price $p$ is the ex-dividend value, so its total gain is $dp+kdt$. The replicating <portfolio> pays the same amounts out of its wealth, with no external injections. Thus its gain identity is
$$
dp+k(S_t,t)dt=\Delta\,dS_t+\rho(p-\Delta S_t)dt.
$$
Again the Brownian coefficients force $\Delta=p_x$. Comparing <drifts> now gives the <Black-Scholes equation with claim dividends>:
$$
\boxed{p_t+\rho xp_x+\tfrac12\sigma^2x^2p_{xx}-\rho p+k(x,t)=0,\qquad p(x,T)=f(x).}
$$
In particular the cash-flow source has a positive sign in the left-hand side. It is not an underlying-stock dividend yield and does not replace the <drift> $\rho x$ by a dividend-adjusted <drift>.
For another justification, under the <risk-neutral measure> $Q$, discount the claim's cum-dividend gain. The <Itô formula> shows that
$$
d\left(e^{-\rho t}p(S_t,t)+\int_0^t e^{-\rho u}k(S_u,u)\,du\right)
=e^{-\rho t}\sigma S_tp_x\,dW_t^Q
$$
exactly when the displayed equation holds. Subject to the usual <integrability> making this a true <martingale>, <conditional expectation> at maturity gives
$$
p(x,t)=\mathbb E_Q\left[e^{-\rho(T-t)}f(S_T)+\int_t^T e^{-\rho(u-t)}k(S_u,u)\,du\,\middle|\,S_t=x\right].
$$
Positive interim payments therefore add positive value, which also checks the source sign. For instance, a constant payment rate $k_0$ and zero terminal payoff have price $k_0(1-e^{-\rho(T-t)})/\rho$, interpreted as $k_0(T-t)$ at $\rho=0$; substitution satisfies $p_t-\rho p+k_0=0$.
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