Instantaneous covariances

The covariance structure of zero-rates $y(t)$ can be obtained right away. With $t\,y_\tau(t)=u^\prime(t)\,\xi_\tau-v(t)$ the conditional zero-rates follow the process

$$\mathrm{d}_\tau y(t) = {1\over t} u^\prime(t)\,\mathrm{d}\xi_\tau \quad ,$$ (14)

where $\Lambda=A^\star-A$, $\lambda^1=a^\star-a$, and the instantaneous conditional covariances are

$$\mathsf{var}\left(\mathrm{d}y(t_1),\mathrm{d}y(t_2)\vert\,\... ...me(t_1)}{t_1}\, V\, \frac{u(t_2)}{t_2}\,\mathrm{d}\tau\quad .$$ (15)

The results of section on eigenvalues of outer product matrices shows thet the covariances $\mathsf{var}\left(\mathrm{d} y(t_1),\mathrm{d}y(t_2)\vert\,\mathcal{F}_\tau\right)$ have exactly $n$ positive eigenvalues (counting multiplicities).