the Lebesgue measure.

The Explicit Euler Scheme: Criteria for Weak and Strong Convergence
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It was shown by Higham, Mao & Stuart in [@HMS] that the explicit Euler scheme $$\begin{aligned}
 \overline{X}_{t_{k+1}} &= \overline{X}_{t_{k}}  + a(\overline{X}_{t_{k}}) \Delta + \sum_{j=1}^m b_j (\overline{X}_{t_{k}}) \Delta_k W^{(j)}, \qquad k=0,1, \ldots,   
\\ \overline{X}_{0} &=x_0  \nonumber \end{aligned}$$ is strongly convergent if the coefficients are locally Lipschitz continuous on ${\mathbb{R}}^d$ and a moment condition for the SDE and its Euler approximation is satisfied. This result can be extended to SDE on domains and the modified or reflected Euler scheme.

\[euler\_hms\_thm\] Let $X$ be the solution of SDE satisfying condition . Moreover, let $ \widetilde{X}^{n}$ be the modified Euler scheme based on the auxiliary functions $f \in C(E; {\mathbb{R}}^d)$, $g \in C(E; {\mathbb{R}}^{d,m}) $ with stepsize $\Delta=T/n$ or let $ \widetilde{X}^{n}$ be the reflected Euler scheme based on the projection function $\psi : E \rightarrow D \cup \partial D$ with stepsize $\Delta=T/n$. Assume that $$a\in C^{ 2 }( D ;
  {\mathbb R}^{d}), \qquad b \in C^{ 2 }( D ;
  {\mathbb R}^{d,m})$$ and furthermore, assume that for some $p >2$ $$\begin{aligned}
 \sup_{n \in \mathbb{N}} \, \mathbf{E} \max_{k=0, \ldots n} |\widetilde{X}^n_{t_k} |^p +  \mathbf{E}   \sup_{t \in [0,T]} |X_t|^p < \infty. 