= Solution
{c}
Use a <self-financing portfolio> with $\Delta_t$ units of the traded asset and the remainder in the <continuous-time bank account>. If its value equals $V(S_t,t)$, its <self-financing condition> is
$$
dV=\Delta_t\,dS_t+r(V-\Delta_tS_t)\,dt.
$$
Meanwhile <Itô's formula> gives
$$
dV=(V_t+\alpha S_tV_S+\tfrac12\sigma^2V_{SS})\,dt+\sigma V_S\,dW_t.
$$
Matching the <Brownian motion> coefficients gives the <delta hedge> $\Delta_t=V_S$. Matching drifts then cancels the physical drift $\alpha$, leaving the <pricing equation for a linear asset with additive noise>:
$$
\boxed{V_t+\frac12\sigma^2V_{SS}+rSV_S-rV=0,\qquad V(S,T)=P_{\mathrm o}(S).}
$$
The domain is the full <real line>: additive noise can take the asset through zero. The <diffusion coefficient> is $\sigma$, so the second-derivative term has no factor $S^2$. As usual, the replication argument assumes suitable differentiability and an <admissible trading strategy>.
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