= Solution
{c}
The differential operator in part <1/b> is the <infinitesimal generator> of the auxiliary <diffusion process> $dS_u=rS_u\,du+\sigma\,dW_u$, with killing rate $r$. The <Feynman-Kac formula> therefore gives
$$
\boxed{V(S,t)=e^{-r(T-t)}\mathbb E^{\mathbb Q}[P_{\mathrm o}(S_T)\mid S_t=S].}
$$
Equivalently, <Itô's formula> makes $e^{-r(u-t)}V(S_u,u)$ a <local martingale> under these dynamics; under appropriate growth and integrability assumptions it is a <martingale>, whose terminal <conditional expectation> yields the same formula. The expectation uses the drift-$r$ law, rather than the physical drift-$\alpha$ law in part <1/a>. This is <risk-neutral pricing>.
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