\big| 
    \mathbb{E}|X_T|^p
    - 
    \mathbb{E}| { \overline{X}}^n_T|^p 
  \big| 
  = \infty .$$

In the case of the 3/2-model, which has finite moments up to order $p < 2+ \frac{2c_1}{c_3^2}$, the coefficients are $$a(x)=- c_2 x^2 + c_1 c_2 x, \qquad b(x)= c_3(x^{+})^{3/2}, \qquad x \in \mathbb{R},$$ so the assumptions of the above Theorem are satisfied for $\alpha=3/2$, $\beta=2$ and $C$ sufficiently large.

Concerning the Ait-Sahalia model, the moments of the Euler scheme already explode in the second step. Here the first step of the Euler scheme has a Gaussian distribution with mean $x_0 + (\alpha_{-1}x_0^{-1} - \alpha_0 + \alpha_1 x_0 - \alpha_2 x_0^{r}) \Delta $ and variance $\alpha_3^2 x_0^{2 \rho}\Delta$. The inverse of the first step must be computed for the second step of the Euler scheme, so the moments of the second step are infinite, since inverse moments of a Gaussian random variable do not exist.

Why the moments of the Euler scheme diverge for superlinearly growing coefficients – even without a singularity – can be nicely illustrated by considering the SDE $$\label{exsde2}
d X_t =
- X_t^3\,dt + \sigma dW_t,
\qquad
X_0 = x_0$$ with $\sigma \geq 0$ for which the Euler scheme reads as $$\label{exeuler2}
{ \overline{X}}^n_{t_{k+1}}
= { 