Dividend-yield discount shift ($q(x,t)=e^{-d(T-t)}p(x,t;r-d)$)

With constant continuous stock-dividend yield $d$, the risk-neutral measure gives stock drift $r-d$ but cash claims are discounted at $r$. The same terminal payoff in a no-dividend Black-Scholes model with interest rate $r-d$ has the identical stock-transition law but a different discount factor. Their prices therefore obey the displayed identity, under the usual payoff integrability conditions.