Consider the so-called Cox-Ingersoll-Process model
\begin{equation} dr_t=a(b-r_t)dt+\sigma \sqrt{r_t}dW_t \end{equation}
It can be shown (Wikipedia), that the mean of this process is
\begin{equation} \langle r_t \rangle = r_0 e^{-at }+ b(1-e^{-at})^2 \end{equation} for some initial value $r_0$.
Consider now the transformation $Y_t=\sqrt{r_t}$. By Itô calculus, it can be shown that this new process fulfills the following stochastic differential equation
\begin{equation} dY_t=\left(\frac{\alpha}{Y_t}-\beta Y_t\right)dt + \gamma dW_t \end{equation}
This transformation is called the Lamperti transformation and is a common way to shift the nonlinearity from the diffusion coefficient into the drift.
For certain reasons, I need to estimate $\langle Y_t \rangle$. Can it be solved?
As a naïve approach, one can assume that $\langle Y_t \rangle=\sqrt{\langle r_t \rangle}$. However, after some numerical simulations, I realized this was wrong.