Cox-Ingersoll-Ross (CIR) model. Consider the equation

\[

\begin{equation*}

d r_{t}=\left(\alpha-\beta r_{t}ight) d t+\sigma \sqrt{r_{t}} d B_{t} \tag{4.45}

\end{equation*}

\]

modeling the variations of a short-term interest rate process \(r_{t}\), where \(\alpha, \beta, \sigma\) and \(r_{0}\) are positive parameters and \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion.

a) Write down the equation (4.45) in integral form.

b) Let \(u(t)=\mathbb{E}\left[r_{t}ight]\). Show, using the integral form of (4.45), that \(u(t)\) satisfies the differential equation

\[

u^{\prime}(t)=\alpha-\beta u(t),

\]

and compute \(\mathbb{E}\left[r_{t}ight]\) for all \(t \geqslant 0\).

c) By an application of Itô's formula to \(r_{t}^{2}\), show that

\[

\begin{equation*}

d r_{t}^{2}=r_{t}\left(2 \alpha+\sigma^{2}-2 \beta r_{t}ight) d t+2 \sigma r_{t}^{3 / 2} d B_{t} \tag{4.46}

\end{equation*}

\]

d) Using the integral form of (4.46), find a differential equation satisfied by \(v(t):=\mathbb{E}\left[r_{t}^{2}ight]\) and compute \(\mathbb{E}\left[r_{t}^{2}ight]\) for all \(t \geqslant 0\).

e) Show that

\[

\operatorname{Var}\left[r_{t}ight]=r_{0} \frac{\sigma^{2}}{\beta}\left(\mathrm{e}^{-\beta t}-\mathrm{e}^{-2 \beta t}ight)+\frac{\alpha \sigma^{2}}{2 \beta^{2}}\left(1-\mathrm{e}^{-\beta t}ight)^{2}, \quad t \geqslant 0

\]