Unconditional finite-time yield dynamics

With the process for the zero-rates given in (), the integration can be carried out to obtain the finite-time differences:

$$\Delta^\delta y_\tau(t) := y_{\tau+\delta}(t)-y_\tau(t) ... ...= {u^\prime(t)\over t}\left(\xi_{\tau+\delta}-\xi_\tau\right)$$ (16)

Unconditional expectation and variance are

$$\begin{eqnarray*} \mathsf{E}\,\Delta^\delta y_\tau(t) &=& 0\\ \mathsf{E}\l... ...1\over\delta}\,\mathsf{E}\left(\Delta^\delta y_\tau(t)\right)^2 \end{eqnarray*}$$

The finite-time covariance $V_\delta$ is given in section as

$$V_\delta = {1\over \delta}\int_0^\delta e^{t\,A}\,V\,e^{t\,A^\prime}\mathrm{d}t \quad .$$ (17)