 Linear bond pricing in a bounded short-rate diffusion

In the model $d{r}_t=r_t(r_t-1)(dt+d{W}_t)$ under the pricing measure, with $0\le r_t\le 1$, the discounted affine expression $D_t[A(T-t)+B(T-t)r_t]$ has zero drift when $A^{\prime }=0$ and $B^{\prime }=-A-B$. Terminal conditions $A(0)=1$, $B(0)=0$ yield the displayed unit bond price. The expression is bounded, so the bounded local martingale criterion justifies pricing by conditional expectation.