Short-rate bond pricing equation
= Short-rate bond pricing equation
{title2=$P_t+\mu_QP_r+\frac12\eta^2P_{rr}-rP=0$}
For pricing drift $\mu_Q$, a unit <zero-coupon bond> price $P(t,r;T)$ satisfies $P_t+\mu_QP_r+\eta^2P_{rr}/2-rP=0$, with $P(T,r;T)=1$. <Itô formula> makes its bank-account-discounted value a local <martingale>; suitable integrability gives $P(t,r;T)=\mathbb E^Q[\exp(-\int_t^T r_sds)\mid r_t=r]$. The rate kills the diffusion semigroup, as in the <Feynman-Kac formula>.