 Deterministic calibration of an autoregressive short rate

For an affine autoregressive process $r_T=\beta r_{T-1}+{\xi }_T$, suppose its initial zero-coupon bond prices $q(T)$ are positive and finite. A deterministic shift $h_T$ of the rate changes the bond price to $q(T)\exp (-{\sum }_{s=1}^T{h}_s)$. To fit any positive curve $p(T)$ with $p(0)=1$, put $d_T=\log q(T)-\log p(T)$, $d_0=0$, $h_T=d_T-d_{T-1}$ and $h_0=0$. Adding the deterministic drift ${\alpha }_T=h_T-\beta h_{T-1}$ to the rate recursion realizes precisely this shift, and telescoping gives the desired curve. No stationary autoregression assumption is needed.