stepsize $\Delta=T/n$. Moreover, let $a_2$ be monotonically decreasing and assume that there exists constants $\alpha \in [0,1/2]$, $\beta \in (0,1]$ and $C>0$ such that $$\begin{aligned}
 |a_1(x)-a_1(y)| \leq C \cdot |x-y|, \qquad  |a_2(x)-a_2(y)| \leq C \cdot |x-y|^{\beta},\end{aligned}$$ $$\begin{aligned}
 |b(x)-b(y)| \leq C \cdot |x-y|^{\frac{1}{2} + \alpha}\end{aligned}$$ for all $x,y \in \mathbb{R}$. Then, for all $p \in \mathbb{N}$, there exist constants $K_p^{\alpha,\beta} >0$ such that $$\mathbf{E}  \max_{k=0, \ldots, n} |X_{t_k}- { \overline{X}}_{t_k}^n |^p  \leq  \left \{ \begin{array}{lcl}K_p^{0,\beta} \cdot \frac{1}{\log(n)} & \textrm{for}&  \alpha=0  \\ K_p^{\alpha,\beta} \cdot \left( \frac{1}{n^{\alpha}} + \frac{1}{n^{\beta/2}} \right) & \textrm{for}&  \alpha \in (0,1/2)\\
K_p^{\alpha,\beta} \cdot \left( \frac{1}{n^{p/2}} + \frac{1}{n^{\beta p/2}} \right) & \textrm{for}&  \alpha =1/2\\
 \end{array} \right.$$

This result can be applied to the CEV model $$dX_t=\mu X_t \, dt + \sigma X_t^{\gamma} \, dW_t,$$ if the mapping $[0, \infty) \ni x \mapsto x^{\gamma} \in [0, \infty) $ is extended to $(-\infty, 0)$, e.g. as $(x^+)^{\gamma}$ or $|x|^{\gamma}$. Theorem \[euler\_hoelder\] then yields strong convergence of the corresponding Euler schemes.

Whether the convergence rates predicted from Theorem \[euler\_hoelder\] are sharp for the CEV model remains an open problem. The following simulation study suggests that the Euler scheme has strong convergence order 1/2, at least for some parameter constellations. To better preserve the 