for negative “excursions". This can also be seen in Figure \[figure\_euler\_exc\], which shows a sample path of the (linearly interpolated) Euler schemes and using the same path of the driving Brownian motion. The parameters used in this figure correspond to Scenario II.

$\diamond$

   

For general SDEs the procedure of modifying the coefficients outside the support of the solution has been introduced systematically in [@JKN]. For an SDE $$\label{snew.sde2}
dX_t = a(X_t)\, dt + \sum_{j=1}^{m} b_{j}(X_t)\,   dW^{(j)}_t,   \qquad X_0=x_0$$ which takes values in a domain $D$ $\subset$ $\mathbb{R}^{d}$, i.e. $$\begin{aligned}
 \mathbf{P}(X_t \in D, \,\, t \geq 0)=1,  \label{support_sde} \end{aligned}$$ the auxiliary coefficients $$\begin{aligned}
\widetilde{a}(x) &=  a(x) \cdot  \mathbf{1}_{D}(x) + f(x) \cdot\mathbf{1}_{E}(x), & \qquad x \in \mathbb{R}^{d}  &
\\[2ex]
\widetilde{b}_j(x)&=  b_j(x) \cdot  \mathbf{1}_{D}(x) + g_j(x) \cdot \mathbf{1}_{E}(x), &\qquad x \in \mathbb{R}^{d}, &\quad j=1, \ldots, m \end{aligned}$$ with $E= \mathbb{R}^d \setminus D$ are introduced there. A modified Itô-Taylor scheme of order $\gamma$ based on the auxiliary functions $f$ and $g$ is then the corresponding standard Itô-Taylor scheme for the SDE $$dX_t = \widetilde{a}(X_t)\, dt + \sum_{j=1}^{m} \widetilde{b}_{j}(X_t)\,  dW^{(j)}_t,$$ with a suitable definition of the derivatives of the coefficients on $\partial D$, see [@JKN] for details. This method is well-defined as long as the 