I am currently doing my thesis on stochastic models applied to interest rates. I am partly basing myself on the article "Stability Behavior of Some Well-Known Stochastic Financial Models" of T. E. Govindan and R. S. Acosta Abreu.
But i'm having trouble understanding the next part (page 1373):
Be \begin{equation} r(t)=e^{-k t}r_{0}+\mu(1-e^{-k t})+\sigma e^{-k t}\int_{0}^{t}e^{k s}\sqrt{r(s)}dW(s) \end{equation}
Applying Ito’s formula to the process $X(t)=e^{k t}r(t)$ gives. \begin{equation} X(t)=r_{0}+\mu(e^{k t}-1)+\sigma\int^{t}_{0}e^{k s}\sqrt{r(s)}dW(s). \end{equation} Taking expectations on both sides, we get \begin{equation} \mathrm{E}X(t)=r_{0}+\mu(e^{k t}-1).........(1) \end{equation} Next, consider the processes $X(t)$ and $X^{*}(t)=e^{k t}r^{*}(t)$, where $r(t)$ and $r^{*}(t)$ are solutions with the initial conditions $r(0)=r_{0}$ and $r^{*}(0)=r^{*}_{0}$ respectively. Exploiting again Ito’s formula, we obtain the following stochastic differential for $\lvert X(t)-X^{*}(t)\lvert^{2}$: \begin{equation} \begin{aligned} &d\lvert X(t)-X^{*}(t)\lvert^{2}=\sigma^{2}e^{k t}(\sqrt{X(t)}-\sqrt{X^{*}(t)})^{2}dt\\ &+2\sigma e^{k t/2}(X(t)-X^{*}(t))(\sqrt{X(t)}-\sqrt{X^{*}(t)})dW(t) \end{aligned} \end{equation} Integrating both sides and then taking expectation, we get \begin{equation*} \mathrm{E}\lvert X(t)-X^{*}(t)\lvert^{2}=\lvert r_{0}-r_{0}^{*}\lvert^{2}+\sigma^{2}\int_{0}^{t}e^{k s}\mathrm{E}(\sqrt{X(s)}-\sqrt{X^{*}(s)})^{2}ds .....(2) \end{equation*} From equations (1) and (2), it follows that \begin{equation} \mathrm{E}\lvert r(t)-r^{*}(t)\lvert^{2}\leq\lvert r_{0}-r_{0}^{*}\lvert^{2}e^{-k t}+\frac{3}{2}(r_{0}+r^{*}_{0})(1-e^{-k t})e^{-k t}+\frac{3\sigma^{2}k}{k}(\frac{1}{2}-e^{-k t}+\frac{1}{2}e^{-2k t}), \hspace{4mm} t\geq 0.....(3) \end{equation}
I can't understand how (3) can be obtained from (1) and (2). I've tried to figure out a way to use Gronwall's lemma but it doesn't seem to work. I hope someone can give me an idea and thanks in advance for taking the time to read.