= Solution
Assume $\sigma>0$, $x>0$ and sufficient $C^{2,1}$ regularity to use the <Itô formula>. Under the physical measure write $dS_t=\mu S_tdt+\sigma S_tdW_t$ and let the <bank account> be $R_t=e^{\rho t}$. If $V_t=f(S_t,t)$ is a <self-financing portfolio> with $\Delta_t$ stock units and $\beta_t$ bank-account units, then
$$
dV_t=\Delta_t\,dS_t+\beta_t\,dR_t,\qquad V_t=\Delta_tS_t+\beta_tR_t.
$$
The <Itô formula> gives $df=(f_t+\mu S_tf_x+\tfrac12\sigma^2S_t^2f_{xx})dt+\sigma S_tf_xdW_t$. Equating diffusion terms forces $\Delta_t=f_x(S_t,t)$; equating drifts and eliminating $\beta_tR_t=f-S_tf_x$ gives the <Black-Scholes equation>
$$
\boxed{f_t+\tfrac12\sigma^2x^2f_{xx}+\rho x f_x-\rho f=0.}
$$
Conversely, if this equation holds, take $\Delta=f_x$ and $\beta=(f-xf_x)/R$. The same comparison shows $df=\Delta\,dS+\beta\,dR$, so the constructed portfolio is <self-financing>. This is a local replication equivalence for smooth functions on the state space; global admissibility and a terminal condition specify a unique economically admissible claim price.
With continuous stock-dividend yield $\theta$, holding $\Delta_t$ stock units earns $\theta\Delta_tS_tdt$ in addition to their price changes. A <self-financing portfolio> reinvests that income, so
$$
dV_t=\Delta_t(dS_t+\theta S_tdt)+\beta_t\,dR_t.
$$
The same diffusion comparison gives $\Delta=f_x$, and the drift comparison yields the <Black-Scholes equation with continuous stock dividends>
$$
\boxed{f_t+\tfrac12\sigma^2x^2f_{xx}+(\rho-\theta)xf_x-\rho f=0.}
$$
Under the <risk-neutral measure>, the ex-dividend stock consequently has drift $\rho-\theta$, while cash is discounted at $\rho$.
For the last claim, distinguish the terminal payoff function $F$ from a time-dependent price. Put $\tau=t_0-t$. The price with yield $\theta$ is $q(x,t)=e^{-\rho\tau}\mathbb E[F(xe^{(\rho-\theta-\sigma^2/2)\tau+\sigma\sqrt\tau Z})]$, with $Z$ a standard <normal random variable>. In a no-dividend model with interest rate $\rho-\theta$, the stock-transition law is identical but its discount factor is $e^{-(\rho-\theta)\tau}$. Thus, whenever the payoff expectation is finite, the <dividend-yield discount shift> gives
$$
\boxed{q(x,t)=e^{-\theta(t_0-t)}p(x,t).}
$$
Equivalently substitute this expression in the dividend equation: $q_t=e^{-\theta\tau}(p_t+\theta p)$, and the equation reduces exactly to the no-dividend <Black-Scholes equation> with rate $\rho-\theta$, with the same terminal data.
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