= Pricing equation for a linear asset with additive noise
{title2=$V_t+rsV_s+\frac12\sigma^2V_{ss}-rV=0$}
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 $V-sV_s$. <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 $dS=rSdt+\sigma dW^{\mathbb Q}$, so the same price is the discounted conditional payoff expectation under the usual admissibility and integrability hypotheses.
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