Pricing equation for a linear asset with additive noise ($V_t+rs{V}_s+\frac{1}{2}{\sigma }^2{V}_{ss}-rV=0$)

For a non-dividend-paying traded asset $dS=\mu Sdt+\sigma dW$ and bank account $dB=rBdt$, delta hedging sets the share holding to $V_s$. Its bond value is $V-s{V}_s$. Itô's formula and the self-financing condition give the displayed pricing equation on the whole real line. The linear drift change of measure supplies risk-neutral dynamics $dS=rSdt+\sigma d{W}^{\mathbb{Q}}$, so the same price is the discounted conditional payoff expectation under the usual admissibility and integrability hypotheses.