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Assume $\sigma >0$, $x>0$ and sufficient $C^{2,1}$ regularity to use the Itô formula. Under the physical measure write $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ 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, thenThe Itô formula gives $df=(f_t+\mu S_t{f}_x+\frac{1}{2}{\sigma }^2{S}_t^2{f}_{xx})dt+\sigma S_t{f}_{x}d{W}_t$. Equating diffusion terms forces ${\Delta }_t=f_x(S_t,t)$; equating drifts and eliminating ${\beta }_t{R}_t=f-S_t{f}_x$ gives the Black-Scholes equationConversely, if this equation holds, take $\Delta =f_x$ and $\beta =(f-x{f}_x)/R$. 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 $\theta {\Delta }_t{S}_{t}dt$ in addition to their price changes. A self-financing portfolio reinvests that income, soThe same diffusion comparison gives $\Delta =f_x$, and the drift comparison yields the Black-Scholes equation with continuous stock dividendsUnder 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 $q(x,t)=e^{-\rho \tau }\mathbb{E}[F(x{e}^{(\rho -\theta -{\sigma }^2/2)\tau +\sigma \sqrt{\tau }Z})]$, 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 givesEquivalently 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.