\mathbb{R}^{d} \rightarrow \mathbb{R}$. The test functions $\phi$ are a particular case of the general (path-dependent) functionals $\Phi$. In the strong approximation problem the $p$-th mean of the difference between $X$ and $\overline{X}$ is analyzed, i.e. $$\Big(\mathbf{E} \sup_{k=0, \ldots, n} | X_{t_{k}} - \overline{X}_{t_{k}} |^{p} \Big)^{1/p}$$ for the maximal error in the discretization points or $$\Big(\mathbf{E} \sup_{t \in [0,T]} | X_{t} - \overline{X}_{t} |^{p} \Big)^{1/p}$$ for the global error, where $p \geq 1$ and $| \cdot |$ denotes the Euclidean norm. Here the mean-square error, i.e. $p=2$, is usually studied. The recent development of the Multi-level Monte Carlo method for SDEs [@g1; @g2] has revealed that strong error bounds are crucial for the efficient computation of functionals $\mathbf{E}\Phi(X)$.

While the strong error measures the error of the approximate sample paths $\overline{X}$ on average, the pathwise error is the random quantity $$\sup_{k=0, \ldots, n}|X_{t_{k}}(\omega) - \overline{X}_{t_{k}}(\omega)|, \qquad \omega \in \Omega$$ and $$\sup_{t \in [0,T] }|X_{t}(\omega) - \overline{X}_{t}(\omega)|, \qquad \omega \in \Omega$$ respectively. Here the error is analyzed for a fixed $\omega \in \Omega$ without averaging. This quantity thus gives the error of the actually calculated approximation $\overline{X}_{t_k}(\omega)$, $k=0, \ldots,n$, respectively $\overline{X}(\omega)$.

The traditional weak and strong convergence analysis for numerical methods for 