1.0, 1.5, \ldots$. For the definition of these schemes, see e.g. [@KP]. The Euler scheme corresponds to $\gamma=0.5$, while $\gamma=1.0$ yields the Milstein scheme $$\begin{aligned}
 \overline{X}_{t_{k+1}} &= \overline{X}_{t_{k}}  + a(\overline{X}_{t_{k}}) \Delta + \sum_{j=1}^m b_j (\overline{X}_{t_{k}}) \Delta_k W^{(j)}  + \sum_{j_1,j_2=1}^m L^{j_1}b_{j_2}(\overline{X}_{t_{k}}) I_{j_1,j_2}(t_k,t_{k+1})\end{aligned}$$ with the differential operators $$L^{j}  =\sum_{k=1}^{d} b_j^{(k)} \frac{\partial}{\partial x^{k}}, \qquad j = 1 , \ldots, m$$ and the iterated Itô-integrals $$I_{j_1,j_2}(s,t) = \int_{s}^{t} 
\int_{s}^{\tau_2} d W^{(j_{1})}_{\tau_{1}} \,  d W^{(j_2)}_{\tau_2}, \qquad j_1,j_2=1, \ldots, m.$$ The Itô-Taylor scheme of order $1.5$ is usually called the Wagner-Platen scheme.

\[thm\_pw\_1\] Let $\gamma$ $=$ $0.5$, $1.0$, $1.5$, $\ldots$. Assume that $a$, $b_1$, $\ldots$, ${b_m}$ $\in$ $C^{ 2 \gamma  +1}(\mathbb{R}^d ;\mathbb{R}^{d})$ and moreover let ${ \overline{X}}^{\gamma ,n}$ be the Itô-Taylor scheme of order $\gamma$ with stepsize $\Delta =T/n$. Then for every $\varepsilon >0$ there exists a non–negative random variable $\eta_{\varepsilon}^{\gamma} $ such that $$\sup_{k=0, \ldots, n}  \left |X_{t_k}(\omega ) -{ \overline{X}}^{\gamma,n}_{t_k}(\omega)
\right| \leq \eta_{ \varepsilon}^{\gamma}(\omega)
\cdot n^{-\gamma + \varepsilon}$$ for almost all $\omega \in \Omega$.

The main ingredients to obtain this result are the Burkholder-Davis-Gundy inequality, which implies that all moments of an Itô-integral are equivalent, the following Borel-Cantelli-type Lemma, and a localization procedure.

(see [@lms]) Let $\alpha >0$, $c_p \geq 0$ for $p \geq 