 Gaussian forward-rate covariance drift restriction

Suppose the instantaneous forward rate field has mean ${\mu }_{s,T}$ and covariance $c(s\wedge r,T,U)$, with deterministic initial curve and suitable integrability and maturity regularity. Its discounted zero-coupon bond prices are martingales exactly when the displayed mean identity holds. PutThe discounted bond price is $\exp [-A(s,T)-Y(s,T)]$, where $Y$ is the integrated Gaussian forward-rate process. Its conditional Gaussian exponential expectation gives the equivalent identity $A(s,T)-A(0,T)=v(s,T)/2$. Differentiating this in maturity gives the displayed restriction. Conversely, integrating that restriction on the diagonal and along the current curve recovers the half-variance identity, by symmetry of the covariance over the maturity square. The Gaussian exponential martingale with deterministic variance then verifies the full conditional martingale property. This formulation does not require a time derivative of $c$.