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Line 618:
_{i}(t_{n})\Delta \mathbf{z}_{n}^{i}+\frac{d\mathbf{g}_{i}(t_{n})}{dt}
(\Delta \mathbf{w}_{n}^{i}h_{n}-\Delta \mathbf{z}_{n}^{i})\right) , (16)</math></small>
where the matrices <math>\mathbf{M}_{n}</math>, <math>\mathbf{L}</math> and <math>\mathbf{r}</math> are defined as
<small><math>\mathbf{M}_{n}=
\begin{bmatrix}
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n}) & \mathbf{f}_{t}(t_{n},\mathbf{
y}_{n})\quad+\frac{1}{2}\sum\limits_{j=1}^{m}\left( \mathbf{I}_{d\times
d}\otimes \mathbf{g}_{j}^{\intercal }\left( t_{n}\right) \right) \mathbf{f}_{
\mathbf{xx}}(t_{n},\mathbf{y}_{n})\mathbf{g}_{j}\left( t_{n}\right) &
\mathbf{f}(t_{n},\mathbf{y}_{n}) \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}
\in \mathbb{R}^{(d+2)\times (d+2)}, </math></small>
<math>\mathbf{L}=\left[
\begin{array}{ll}
\mathbf{I}_{d} & \mathbf{0}_{d\times 2}
\end{array}
\right] , \mathbf{r}^{\intercal }=\left[
\begin{array}{ll}
\mathbf{0}_{1\times (d+1)} & 1
\end{array}
\right]</math>, <math>\Delta \mathbf{z}_{n}^{i}</math> is a i.i.d. zero mean Gaussian random variable with variance <math>E\left( (\Delta \mathbf{z}_{n}^{i})^{2}\right) =%
\frac{1}{3}h_{n}^{3}</math> and covariance <math>E(\Delta \mathbf{w}_{n}^{i}\Delta
\mathbf{z}_{n}^{i})=\frac{1}{2}h_{n}^{2}</math> and '''p+q>1.''' For large systems of SDEs, in the above scheme <math>(\mathbf{P}_{p,q}(2^{-k_{n}}\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r}</math> is replaced by <math>\mathbf{\mathbf{k}}
_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}_{n}^{\gamma },\mathbf{r})</math>.
==== Order 2 SLL-Taylor schemes ====
<small><math>\mathbf{y}_{t_{n+1}} =\mathbf{y}_{n}+\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}
\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r}+\sum\limits_{j=1}^{m}\mathbf{g}
_{j}\left( t_{n}\right) \Delta \mathbf{w}_{n}^{j}+\sum\limits_{j=1}^{m}
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\mathbf{g}_{j}\left(
t_{n}\right) \widetilde{J}_{\left( j,0\right) }
+\sum\limits_{j=1}^{m}\frac{d\mathbf{g}_{_{j}}}{dt}\left( t_{n}\right)
\widetilde{J}_{\left( 0,j\right) }+\sum\limits_{j_{1},j_{2}=1}^{m}\left(
\mathbf{I}_{d\times d}\otimes \mathbf{g}_{j_{2}}^{\intercal }\left(
t_{n}\right) \right) \mathbf{f}_{\mathbf{xx}}(t_{n},\mathbf{y}_{n})\mathbf{g}
_{j_{1}}\left( t_{n}\right) \widetilde{J}_{\left( j_{1},j_{2},0\right) },</math></small>
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