\big | \leq \eta_{\gamma, \varepsilon}^{f,g}(\omega)
\cdot n^{-\gamma + \varepsilon}$$ for almost all $\omega \in \Omega$ and all $n \in {\mathbb N}$.

In the case of the Euler scheme, i.e. $\gamma =0.5$, the assumptions on $a$ and $b$ can be weakened to the assumption that $a$ and $b$ are locally Lipschitz continuous on $D$. For SDEs on domains in mathematical finance this condition is typically satisfied. In fact, in most cases the coefficients are infinitely differentiable.

The CIR process satisfies $$\mathbf{P}(X_t > 0 \,\,\, {\textrm{for all}} \, \,\, t \geq 0)=1$$ if and only if $2 \kappa \lambda \geq \theta^2$. The latter assumption is typically satisfied in interest rate applications of the CIR process. Hence, modified Taylor schemes can be used here with $D=(0, \infty)$. The truncated Euler scheme (\[euler\_2\]) corresponds to the auxiliary functions $f(x)=a(x)$, $g(x)=0$, $x \leq 0$, while the scheme (\[euler\_3\]) uses the auxiliary functions $f(x)=a(x)$, $g(x)=\sqrt{-x}$, $x \leq 0$. Note that $2\kappa \lambda \geq \theta^2$ is satisfied in Scenario I, but not in Scenario II.

For structure preserving integration of the CIR process also the symmetrized Euler method $$\begin{aligned}
\label{euler_1}
\widetilde{X}_{t_{k+1}} &= \Big | \widetilde{X}_{t_k} +  \kappa( \lambda
  - \widetilde{X}{t_k} )  \Delta  + \theta  \sqrt{
     \widetilde{X}_{t_k}} \, 