= Gaussian forward-rate covariance drift restriction
{c}
{title2=$\mu_{s,T}=\mu_{0,T}+\int_0^Tc(s\wedge u,u,T)du$}
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. Put
$$
A(s,T)=\int_0^s\mu_{u,u}du+\int_s^T\mu_{s,u}du.
$$
The 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 $c$.
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