\Delta_k W \Big|, \qquad k=0,1, \ldots  \end{aligned}$$ was proposed in [@diop; @diop_2]. While the modified Euler schemes and may leave $(0, \infty) $ and are then forced back in the next steps, this scheme is always non-negative. Adapting this to general SDEs, which take values in a domain $D$, leads to the reflected Euler schemes, see e.g. [@NZ], which are given by $$\begin{aligned}
 \widetilde{X}^{\psi}_{t_{k+1}}  =   H^{\psi}_{t_{k+1}}  \cdot \mathbf{1}_{D}( H^{\psi}_{t_{k+1}}) + \psi(H^{\psi}_{t_{k+1}})  \cdot \mathbf{1}_{\mathbb{R}^d \setminus D}( H^{\psi}_{t_{k+1}}) \end{aligned}$$ with $ \widetilde{X}^{\psi }_{0} = x_{0},$ where $$\begin{aligned}
H^{\psi}_{t_{k+1}} = \widetilde{X}^{\psi}_{t_{k}} +  a(\widetilde{X}^{\psi}_{t_{k}}) \Delta + \sum_{j=1}^m b_j (\widetilde{X}^{\psi}_{t_{k}}) \Delta_k W^{(j)}\end{aligned}$$ and a measurable projection function $\psi: \mathbb{R}^d \setminus D \rightarrow D \cup \partial D$. A straightforward modification of the above theorem yields a pathwise convergence order $1/2-\varepsilon$ for these reflected Euler schemes if the SDE coefficients are twice continuously differentiable on $D$. In the same way reflected Itô-Taylor schemes of arbitrary order can be constructed and analyzed.

The symmetrized Euler scheme corresponds to the reflection function $\psi(x)=|x|$. The results on modified Itô-Taylor schemes and reflected Euler methods apply also to the generalized Ait-Sahalia model with $D=(0,\infty)$ if $r>1, \rho < (1+r)/2$, to the Heston model with $D=(0,\infty)^2$ if $2 \kappa \lambda \geq \theta^2$ 