(Degenerate two-factor Vasicek model). In the discussion of short rates and long rates in the two-factor Vasicek model of Subsection 10.2.1, we made the assumptions that $\delta_2 \neq 0$ and $\left(\lambda_1-\lambda_2\right) \delta_1+\lambda_{21} \delta_2 \neq 0$ (see Lemma 10.2.2). In this exercise, we show that if either of these conditions is violated, the two-factor Vasicek model reduces to a one-factor model, for which long rates and short rates are perfectly correlated (see Exercise 10.5).
(i) Show that if $\delta_2=0$ (and $\delta_0>0, \delta_1>0$ ), then the short rate $R(t)$ given by the system of equations (10.2.4)-(10.2.6) satisfies the one-dimensional stochastic differential equation
$$
d R(t)=(a-b R(t)) d t+d \widetilde{W}_1(t) .
$$
Define $a$ and $b$ in terms of the parameters in (10.2.4)-(10.2.6).
(ii) Show that if $\left(\lambda_1-\lambda_2\right) \delta_1+\lambda_{21} \delta_2=0$ (and $\delta_0>0, \delta_1^2+\delta_2^2 \neq 0$ ), then the short rate $R(t)$ given by the system of equations (10.2.4)-(10.2.6) satisfies the one-dimensional stochastic differential equation
$$
d R(t)=(a-b R(t)) d t+\sigma d \tilde{B}(t) .
$$
Define $a$ and $b$ in terms of the parameters in (10.2.4)-(10.2.6) and define the Brownian motion $\widetilde{B}(t)$ in terms of the independent Brownian motions $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ in $(10.2 .4)$ and $(10.2 .5)$.
10.7 Exercises
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