(Calibration of the two-factor Vasicek model). Consider the canonical two-factor Vasicek model (10.2.4), (10.2.5), but replace the interest rate equation $(10.2 .6)$ by
$$
R(t)=\delta_0(t)+\delta_1 Y_1(t)+\delta_2 Y_2(t)
$$
where $\delta_1$ and $\delta_2$ are constant but $\delta_0(t)$ is a nonrandom function of time. Assume that for each $T$ there is a zero-coupon bond maturing at time $T$. The price of this bond at time $t \in[0, T]$ is
$$
B(t, T)=\tilde{\mathbb{E}}\left[e^{-\int_t^T R(u) d u} \mid \mathcal{F}(t)\right] .
$$
Because the pair of processes $\left(Y_1(t), Y_2(t)\right)$ is Markov, there must exist some function $f\left(t, T, y_1, y_2\right)$ such that $B(t, T)=f\left(t, T, Y_1(t), Y_2(t)\right)$. (We indicate the dependence of $f$ on the maturity $T$ because, unlike in Subsection 10.2.1, here we shall consider more than one value of $T$.)
(i) The function $f\left(t, T, y_1, y_2\right)$ is of the affine-yield form
$$
f\left(t, T, y_1, y_2\right)=e^{-y_1 C_1(t, T)-y_2 C_2(t, T)-A(t, T)} .
$$
Holding $T$ fixed, derive a system of ordinary differential equations for $\frac{d}{d t} C_1(t, T), \frac{d}{d t} C_2(t, T)$, and $\frac{d}{d t} A(t, T)$.