W^{(j)}, \qquad k=0,1, \ldots,    \end{aligned}$$ with $\overline{X}_{0}  =x_0$, where $\Delta =T/n$, $t_k=k\Delta$ and $\Delta_k W = W_{t_{k+1}}-W_{t_k}$. The Euler scheme (and all other approximation methods that will be introduced below) depend on the stepsize $\Delta>0$, hence on $n \in \mathbb{N}$, but this dependence will be omitted whenever it is clear from the context.

From the results of Gyöngy [@gyoengy] it follows that the Euler scheme has pathwise convergence order $1/2-\varepsilon$ also if the SDE coefficients are only locally Lipschitz continuous: for all $\varepsilon >0 $ $$\sup_{k=0, \ldots, n}|X_{t_{k}} - \overline{X}_{t_{k}}| \leq \eta_{{\varepsilon}}^{E}  \cdot n^{-1/2 + \varepsilon}$$ almost surely for a finite and non-negative random variable $\eta_{\varepsilon}^E$ under the assumption that for all $N \in \mathbb{N}$ there exist constants $L_N>0$ such that $$\begin{aligned}
 |a(x)-a(y)| + \sum_{j=1}^m |b_{j}(x)-b_j(y)| \leq L_N \cdot |x-y|, \qquad |x|,|y| \leq N.             \end{aligned}$$ Thus, the pathwise convergence rate of the Euler scheme coincides up to an arbitrarily small $\varepsilon >0$ with its strong convergence rate $1/2$, but for the pathwise convergence rate no global Lipschitz assumption is required.

Jentzen, Kloeden & Neuenkirch [@JKN] observed that this is not a specific feature of the Euler scheme but, in fact, holds for general Itô-Taylor schemes of order $\gamma = 
0.5, 