= Solution
Write $T=t_0$ and $B_t=e^{-\rho(T-t)}$ for the risk-free asset paying one at $T$. Under the physical measure, $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$, with $\sigma>0$, while $dB_t=\rho B_t\,dt$. A <self-financing portfolio> with predictable <stock> and bond holdings $g_t,h_t$ has value $V_t=g_tS_t+h_tB_t$ and gains
$$
dV_t=g_t\,dS_t+h_t\,dB_t.
$$
Rebalancing therefore adds no outside money and withdraws none. This gains equation, rather than merely the value decomposition, defines <self-financing>.
Let $p(x,t)=xg(x,t)+B_th(x,t)$. The <Itô formula> gives
$$
dp(S_t,t)=\left(p_t+\mu S_t p_x+\frac12\sigma^2S_t^2p_{xx}\right)dt+\sigma S_t p_x\,dW_t.
$$
The prescribed <portfolio> gains instead have diffusion coefficient $\sigma S_tg$ and drift $\mu S_tg+\rho B_th$. Equality of diffusion coefficients requires $p_x=g$, equivalently
$$
\boxed{xg_x+B_th_x=0.}
$$
Under this condition, $p_{xx}=g_x$. Equality of drifts is then $p_t+\frac12\sigma^2x^2g_x=\rho B_th$. Since $p_t=xg_t+B_th_t+\rho B_th$, this becomes
$$
\boxed{\frac12\sigma^2x^2g_x+xg_t+B_th_t=0.}
$$
These conditions are also sufficient by the same <Itô formula>. Necessity initially holds along the <stock> process. Its positive transition density on $(0,\infty)$ for every positive time, together with continuity of the coefficient functions, extends the identities to all $x>0$ and then to the time endpoints by continuity.
For a <self-financing portfolio>, substitute $g=p_x$ and $B_th=p-xp_x$ into the drift identity to obtain the <Black-Scholes equation>
$$
\boxed{p_t+\frac12\sigma^2x^2p_{xx}+\rho xp_x-\rho p=0.}
$$
Conversely, a smooth solution of this equation gives a <self-financing portfolio> by choosing the <delta hedge>
$$
\boxed{g=p_x,\qquad h=\frac{p-xp_x}{B_t}.}
$$
Its value is $p$, and both coefficient identities follow directly.
The printed deduction needs this holdings qualification: the <Black-Scholes value equation needs delta-compatible holdings>. A value function alone cannot make an arbitrary prescribed decomposition self-financing. As a counterexample, take $p=0$, $g=1$, $h=-x/B_t$. The value function solves the <Black-Scholes equation>, but this zero-value <portfolio> has gains $dS_t-\rho S_t\,dt$, which contain the nonzero term $\sigma S_t\,dW_t$. Thus for the originally specified $g,h$, the exact equivalence is the pricing equation together with $g=p_x$, not the pricing equation alone.
For a claim paying the holder at rate $k(S_t)$, let $p$ denote its remaining, ex-dividend value. A replicating <portfolio> pays out $k(S_t)dt$, so its net wealth equation is $dp=g\,dS+h\,dB-k(S_t)dt$. The <delta hedge> is still $g=p_x$, but the <Black-Scholes equation> becomes
$$
\boxed{p_t+\frac12\sigma^2x^2p_{xx}+\rho xp_x-\rho p+k(x)=0.}
$$
If there is a terminal payoff $f$, set $p(x,T)=f(x)$; with no terminal payment set it to zero. The corresponding <risk-neutral valuation> is
$$
p(x,t)=\mathbb E_Q\left[e^{-\rho(T-t)}f(S_T)+\int_t^T e^{-\rho(u-t)}k(S_u)\,du\ \middle|\ S_t=x\right].
$$
Each intervening payment is discounted from its own payment time.
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