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The differential operator in part b is the infinitesimal generator of the auxiliary diffusion process $d{S}_u=r{S}_{u}du+\sigma d{W}_u$, with killing rate $r$. The Feynman-Kac formula therefore givesEquivalently, 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 a. This is risk-neutral pricing.

Apply part a with $\alpha =r$. For $\tau >0$, the transition density is the normal distribution with mean $m=S{e}^{r\tau }$ and positive variance $v={\sigma }^2(e^{2r\tau }-1)/(2r)$. Integrating the financial payoff against this probability density function givesThus $d_{\ast }=(x-S{e}^{r\tau })/\sqrt{v}$, as required. The variance is positive even if $r<0$, because numerator and denominator have the same sign. For $r=0$, replace $m$ by $S$ and $v$ by ${\sigma }^2\tau$; at expiry use the terminal value directly. The integral presumes an integrable financial payoff.