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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 isMeanwhile Itô's formula givesMatching 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: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.

For $V\in C^{1,2}$, Itô's formula givesMatch the Brownian coefficient with the self-financing portfolio by choosing the option delta $\varphi =V_s$ and bond value $\psi B=V-s{V}_s$. Equating drifts cancels the physical drift $\mu$ and givesThis is the pricing equation for a linear asset with additive noise. Its spatial domain is the whole real line, because the asset can be negative. In particular the diffusion term is ${\sigma }^2{V}_{ss}/2$, not ${\sigma }^2{s}^2{V}_{ss}/2$. The derivation is delta hedging and does not impose a boundary condition at $s=0$.