\overline{X}}^n_{t_k} \left(1 -  |{ \overline{X}}^n_{t_k}|^2 \Delta \right)
+ \sigma \Delta_k W.$$

In the deterministic case, i.e., and with $\sigma=0$, the Euler approximation of the deterministic equation is known to be unstable if the initial value is large (see e.g. Chapter 6 in [@db02]). For example, if $x_0=n$, $T=1$ and $\Delta = n^{-1}$ then $${ \overline{X}}_{t_1}^n= n \left(1- \frac{n^2}{n}\right) 
\approx-n^2$$ and therefore $${ \overline{X}}_{t_2}^n = { \overline{X}}_{t_1}^n \left(1 -  |{ \overline{X}}^n_{t_1}|^2 \Delta \right)
\approx n^5.$$ Iterating this further, one obtains $$\left| { \overline{X}}^n_{t_k} \right| 
\gtrapprox n^{ \left( 2^k \right) }$$ for $k=0,1,\dots,n$. Thus, ${ \overline{X}}_{t_n}^n$ grows double-exponentially fast in $n$. In the presence of noise ($\sigma>0$) there is an exponentially small event that the Brownian motion leaves the interval $[-2 n,2 n]$ and on this event the approximations grow double-exponentially fast due to the deterministic dynamics. Consequently this double-exponentially growth can not be compensated by the exponentially small probability of this event, which leads to the moment explosion of the Euler approximation.

That rare events lead to the explosion of the moments of the Euler scheme can be also seen from the following numerical example. Consider the volatility process in the 3/2-model $$dV_t =  c_1 V_t( c_2 -  V_t) \, dt 