I'm trying to calculate $E[\sqrt{V_t}]$, $V_t$ following a CIR process :
$dV_t = \kappa(\theta - V_t)dt + \xi \sqrt{V_t}dW_t$.
I've used the Ito lemma to compute:
$d\sqrt{V_t} = \bigg(0+\kappa(\theta - V_t)\frac{1}{2\sqrt{V_t}} + \frac{\xi^2V_t}{2}\frac{-1}{4V_t\sqrt{V_t}} \bigg)dt + \xi \sqrt{V_t}\frac{1}{2\sqrt{V_t}}dW_t$
And tried solving the deterministic part - assuming the second part is of null expected value and variance $\frac{\xi^2t}{4}$.
Let $X_t = E[\sqrt{V_t}]$ :
$\frac{\partial X_t^2}{\partial t}=2X_t\frac{\partial X_t}{\partial t} = \kappa(\theta - X_t^2) - \frac{\xi^2}{4}$
This has a closed form solution.
However, simulating the process via Monte Carlo and Finite Differences does not yield the same solution. Am I doing anything wrong, or is there a method to solve it ?
Thanks.