Integrated Gaussian forward-rate process ($Y(s,T)={\int }_0^{T}X(s\wedge u,u)du$)

For a centered Gaussian forward-rate field with covariance $\mathrm{Cov}(X(s,u),X(r,w))=c(s\wedge r,u,w)$, integrate the short-rate past and the current forward curve together using the displayed mean-square integral. Its variance isFor fixed $T$, its increments are independent of the full earlier forward-rate filtration: their covariance with $X(z,w)$ at $z\le r\le s$ is the integral of $c(s\wedge u\wedge z,u,w)-c(r\wedge u\wedge z,u,w)=0$. Use the uncorrelated jointly Gaussian variables are independent principle first for finite collections and then for their generated sigma-algebra. With $c(0,u,w)=0$, this centered process starts at zero.

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 $c$.