= Integrated Gaussian forward-rate process
{title2=$Y(s,T)=\int_0^T X(s\wedge u,u)du$}
For a centered <Gaussian forward-rate field> with covariance $\operatorname{Cov}(X(s,u),X(r,w))=c(s\wedge r,u,w)$, integrate the short-rate past and the current forward curve together using the displayed <mean-square integral>. Its <variance> is
$$
v(s,T)=\int_0^T\int_0^T c(s\wedge u\wedge w,u,w)\,du\,dw.
$$
For fixed $T$, its increments are independent of the full earlier forward-rate <filtration>: their covariance with $X(z,w)$ at $z\leq r\leq s$ 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.
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