Consider the [102] model

\[\begin{aligned}d S_{t} & =\mu_{S} S_{t} d t+\sqrt{Y_{t}} S_{t} d B_{t} \\d Y_{t} & =\mu_{Y} Y_{t} d t+\xi Y_{t} d W_{t}\end{aligned}\]

The volatility process is \(\sqrt{Y_{t}}\). Show that the volatility process has moments

\[\begin{aligned}& \mathbf{E}\left[\sqrt{Y_{t}}\right]=\sqrt{Y_{0}} e^{\frac{1}{2} \mu_{Y} t-\frac{1}{8} \xi^{2} t} \\& V\left[\sqrt{Y_{t}}\right]={\sqrt{Y_{0}}}^{2} e^{\mu_{Y} t}\left(1-e^{-\frac{1}{4} \xi^{2} t}\right)\end{aligned}\]

and study what happens with these moments when \(t \rightarrow \infty\) depending on the values of the parameters in the model.