 Forward-rate equation for a Markov short-rate diffusion

Suppose $d{r}_t=a(r_t)dt+b(r_t)d{W}_t$ under a risk-neutral measure and $f_t(T)=F(T-t,r_t)$. By Itô formula, the forward-rate volatility is ${\sigma }_t(T)=b(r_t)F_r(T-t,r_t)$. The Heath-Jarrow-Morton model drift restriction is equivalent toTo verify it directly for bonds, put $G(\theta ,r)={\int }_0^{\theta }F(s,r)ds$. Integration gives $F-r=a{G}_r+\frac{1}{2}{b}^2{G}_{rr}-\frac{1}{2}{b}^2{G}_r^2$. The zero-coupon bond price $P(t,T)=e^{-G(T-t,r_t)}$ therefore satisfiesIts bank-account-discounted price is a positive local martingale. A common equivalent local martingale measure excludes admissible arbitrage for any finite collection of these traded maturities.