+  c_3 V_t^{3/2}  \, dW_t,  \quad V_0=v_0>  0$$ with $$c_1=1.2, \qquad  c_2=0.8, \qquad  c_3=1, \qquad T=4, \qquad v_0= 0.5$$ and try to compute $${\mathbf E} |X_T|=  0.566217$$ using the standard Monte Carlo estimator $$\frac{1}{N} \sum_{i=1}^{N} |{ \overline{X}}_{T}^{n,(i)}|$$ where ${ \overline{X}}_{T}^{n,(1)}, \ldots, { \overline{X}}_{T}^{n,(N)}$ are iid copies of ${ \overline{X}}_{T}^{n}$. The exact value for ${\mathbf E} |X_T|$ is computed using the inverse moments of the CIR process, see e.g. [@HK]. While for a moderate number of repetitions the estimator seems to converge for small step sizes (and the ’Inf’-outputs seem to be some numerical instabilities due to the large step sizes), the estimator explodes even for small step sizes when increasing the number of repetitions – as predicted by Theorem \[ref\_euler\_mom\_exp\]. Despite of this the Euler scheme for this SDE converges pathwise with rate $1/2-\varepsilon$ due to Theorem \[taylor\_mod\].

   Repetitions $N$ / stepsize $\Delta$    $2^0$     $ 2^{-2}$   $ 2^{-4 }$   $2^{-6}$   $ 2^{-8}$   $2^{-10}$
  ------------------------------------- ---------- ----------- ------------ ---------- ----------- -----------
                 $10^3$                  6.327232      Inf         Inf       0.550185   0.553499    0.555069
                 $10^4$                  6.894698      Inf         Inf         Inf      0.562716    0.563352
                 $10^5$                  7.430606      Inf         Inf         Inf      0.566218    0.567106
                 $10^6$                  7.227379      Inf         Inf         Inf         Inf      0.565750
                 $10^7$                  7.279187      Inf         Inf         Inf         Inf         Inf

A similar moment explosion arises 