Quadratic Ornstein-Uhlenbeck state-price density (${\zeta }_t=e^{-\alpha t}(a+X_t^2/2)$)

For an Ornstein-Uhlenbeck process $dX=\sigma dB-\lambda Xdt$ under a reference measure, this is a positive supermartingale when $\alpha a\ge {\sigma }^2/2$. Its short rate is $[\alpha a-{\sigma }^2/2+(\lambda +\alpha /2)X^2]/(a+X^2/2)$. Conditional zero-coupon bond prices are ratios of the OU second moments of the quadratic factor. The money-market density has diffusion coefficient $\sigma X/(a+X^2/2)$, bounded by $\sigma /\sqrt{2a}$, so the Novikov condition validates the pricing measure. The reference OU dynamics must not be silently treated as the risk-neutral dynamics.