4. Hull and White [89] propose the two-factor model
$$
\begin{aligned}
& d U(t)=-\lambda_1 U(t) d t+\sigma_1 d \widetilde{B}_2(t), \\
& d R(t)=\left[\theta(t)+U(t)-\lambda_2 R(t)\right] d t+\sigma_2 d \widetilde{B}_1(t),
\end{aligned}
$$
where $\lambda_1, \lambda_2, \sigma_1$, and $\sigma_2$ are positive constants, $\theta(t)$ is a nonrandom function, and $\widetilde{B}_1(t)$ and $\widetilde{B}_2(t)$ are correlated Brownian motions with $d \widetilde{B}_1(t) d \widetilde{B}_2(t)=$ $\rho d t$ for some $\rho \in(-1,1)$. In this exercise, we discuss how to reduce this to the two-factor Vasicek model of Subsection 10.2.1, except that, instead of (10.2.6), the interest rate is given by $(10.7 .7)$, in which $\delta_0(t)$ is a nonrandom function of time.
10.7 Exercises
455
(i) Define
$$
\begin{gathered}
X(t)=\left[\begin{array}{l}
U(t) \\
R(t)
\end{array}\right], \quad K=\left[\begin{array}{cc}
\lambda_1 & 0 \\
-1 & \lambda_2
\end{array}\right], \quad \Sigma=\left[\begin{array}{cc}
\sigma_1 & 0 \\
0 & \sigma_2
\end{array}\right] \\
\Theta(t)=\left[\begin{array}{c}
0 \\
\theta(t)
\end{array}\right], \quad \widetilde{B}(t)=\left[\begin{array}{c}
\widetilde{B}_1(t) \\
\widetilde{B}_2(t)
\end{array}\right],
\end{gathered}
$$
so that $(10.7 .10)$ and $(10.7 .11)$ can be written in vector notation as
$$
d X(t)=\Theta(t) d t-K X(t) d t+\Sigma d \widetilde{B}(t) .
$$
Now set
$$
\widehat{X}(t)=X(t)-e^{-K t} \int_0^t e^{K u} \theta(u) d u .
$$
Show that
$$
d \widehat{X}(t)=-K \widehat{X}(t) d t+\Sigma d \widetilde{B}(t)
$$
(ii) With
$$
C=\left[\begin{array}{cc}
\frac{1}{\sigma_1} & 0 \\
-\frac{\rho}{\sigma_1 \sqrt{1-\rho^2}} & \frac{1}{\sigma_2 \sqrt{1-\rho^2}}
\end{array}\right],
$$
define $Y(t)=C \widehat{X}(t), \widetilde{W}(t)=C \Sigma \widetilde{B}(t)$. Show that the components of $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are independent Brownian motions and
$$
d Y(t)=-\Lambda Y(t)+d \widetilde{W}(t),
$$
where
$$
\Lambda=C K C^{-1}=\left[\begin{array}{cc}
\lambda_1 & 0 \\
\frac{\rho \sigma_2\left(\lambda_2-\lambda_1\right)-\sigma_1}{\sigma_2 \sqrt{1-\rho^2}} & \lambda_2 .
\end{array}\right] .
$$
Equation (10.7.14) is the vector form of the canonical two-factor Vasicek equations (10.2.4) and (10.2.5).
(iii) Obtain a formula for $R(t)$ of the form (10.7.7). What are $\delta_0(t), \delta_1$, and $\delta_2$ ?