coefficients of the equation are $(2\gamma+1)$-times differentiable on $D$ and the auxiliary functions are $(2\gamma-1)$-times differentiable on $E$. The purpose of the auxiliary functions is twofold: to obtain a well-defined approximation scheme and to bring the numerical scheme back to $D$ if it leaves $D$. In particular, the auxiliary functions can always be chosen to be affine or even constant. It was shown by Jentzen, Kloeden & Neuenkirch [@JKN] that Theorem \[thm\_pw\_1\] adapts to modified Itô-Taylor schemes for SDEs on domains $ D \subset \mathbb{R}^{d}$.

\[taylor\_mod\] Let $X$ be the solution of SDE satisfying condition . Moreover let $\gamma = 0.5, 1.0, 1.5, \ldots$ and assume that $$a\in C^{ 2   \gamma  +1 }( D ;
  {\mathbb R}^{d}), \qquad b \in C^{ 2   \gamma +1 }( D ;
  {\mathbb R}^{d,m})$$ and $$f \in    C^{2
  \gamma  -1 }( E ;
  {\mathbb R}^{d} ), \qquad g \in   C^{       2
 \gamma  -1}( E ;
  {\mathbb R}^{d,m}) .$$ Finally let $ \widetilde{X}^{\gamma,n}$ be the modified Itô-Taylor method for $X$ based on the auxiliary functions $f$ and $g$ with stepsize $\Delta=T/n$. Then for every $\varepsilon >0$ there exists a finite and non-negative random variable $\eta_{\gamma, \varepsilon}^{f,g} $ such that $$\sup_{k=0, \ldots, n}  \big |X_{t_k}(\omega ) -\widetilde{X}^{\gamma,n}_{t_k}(\omega)
