positivity of the CEV process, the Euler scheme is applied to the SDE $$dX_t= \mu |X_t| \, dt + \sigma (X_t^+)^{\gamma} \, dW_t$$ which still fulfills the assumptions of Theorem \[euler\_hoelder\] with $a_2=0$, i.e. $\beta=1$, and $\alpha=\gamma-1/2$. Its solution coincides with the CEV process.

Figure \[figure\_cev\_conv-rate\] shows the empirical root mean square maximum error in the discretization points versus the step size for the parameters $$\begin{aligned}
\textrm{\,Set I:} &\qquad \mu=0.1, \quad \sigma =0.3,  \quad \gamma=0.75, \quad T=1, \quad x_0=0.2\\
\textrm{Set II:} &  \qquad \mu=0.2, \quad  \sigma =0.5, \quad \gamma =0.55, \quad T=1, \quad x_0=0.5\end{aligned}$$ The empirical mean square maximum error in the discretization points is estimated by $$\left( \frac{1}{N} \sum_{i=1}^N    \max_{k=0, \ldots, n} |X_{t_k}^{*,(i)}- { \overline{X}}_{t_k}^{n,(i)} |^2\right)^{1/2}$$ with $N=5 \cdot 10^4$. Here $X^{*}$ is the numerical reference solution obtained by using the same Euler scheme with very small step size and $X^{*,(i)}, { \overline{X}}^{n,(i)}$ are independent copies of $X^{*}, { \overline{X}}^{n}$. For both sets of parameter a good accordance with the convergence order $1/2$ is obtained. (The dots in the figure indicate convergence order $1/2$).

A regression of the numerical data yields moreover the empirical convergence order 0.493923 for set I, respectively 0.509903 for set II.

$\diamond$

But do Theorems 