the $\gamma$-Itô-Taylor schemes, so the pathwise convergence rates carry over.

Note that all the examples of SDEs given in the introduction take non-negative values only, so good approximation schemes should preserve this structural property. The (explicit) Euler scheme is, in general, not such a scheme, since its increments are conditionally Gaussian. For example, in case of the CIR process $$dX_t= \kappa(\lambda- X_t) \,d t +  \theta \sqrt{X_t} \, dW_t, \qquad X_0=x_0 >0$$ the transition density of the Euler scheme reads as $$p(y;x)= \frac{1}{  \sqrt{2 \pi \theta^2 x \Delta}} \exp \left( -\frac{\big(y -( x +\kappa(\lambda -x) \Delta \big)^2}{2 \theta^2 x \Delta}\right), \qquad y \in \mathbb{R}, \, x >0,$$ so negative values can be obtained with positive probability even in the first step. This has lead to many ad-hoc corrections to prevent termination of the Euler scheme. The truncated Euler scheme $$\begin{aligned}
 \label{euler_2}
{ \overline{X}}_{t_{k+1}} &= { \overline{X}}_{t_k} +  \kappa( \lambda - { \overline{X}}_{t_k} ) \, \Delta + \theta  \sqrt{ { \overline{X}}_{t_k}^+ } \, \Delta_k W  , \qquad k=0, 1, \ldots\end{aligned}$$ was proposed in [@delbaen], while the scheme $$\begin{aligned}
 \label{euler_3}
{ \overline{X}}_{t_{k+1}} &= { \overline{X}}_{t_k} +  \kappa( \lambda- { \overline{X}}_{t_k} )  \Delta + \theta  \sqrt{ |{ \overline{X}}_{t_k}| } \, \Delta_k W  , 