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\mathbf{0}_{1\times (d+1)} & 1
\end{array}
\right]</math>, and '''''p+q>1'''''. For large systems of RDEs,
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L\mathbf{k}}
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== Strong LL methods for SDEs ==
Consider the ''d''-dimensional [[Stochastic differential equation|Stochastic Differential Equation]] (SDE)
<math>d\mathbf{x}(t)=\mathbf{f}(t,\mathbf{x}(t))dt+\sum\limits_{i=1}^{m}\mathbf{g}
_{i}(t)d\mathbf{w}^{i}(t),\quad t\in \left[ t_{0},T\right] , \qquad \qquad \qquad (14)</math>
with initial condition <math>\mathbf{x}(t_{0})=\mathbf{x}_{0}</math>, where the drift coefficient <math>\mathbf{f}</math> and the diffusion coefficient <math>\mathbf{g}_{i}</math> are differentiable functions, and <math>\mathbf{w=(\mathbf{w}}^{1},\ldots ,\mathbf{w}
^{m}\mathbf{)}</math> is an ''m''-dimensional standard Wiener process.
=== Local Linear discretization ===
For a time discretization <math>\left( t\right) _{h}</math> , the order-<math>\mathbb{\gamma }</math> (=1,1.5) ''Strong Local Linear discretization'' of the solution of
the SDE (14) is defined by the recursive relation.
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }_{\mathbb{\gamma }}(t_{n},
\mathbf{z}_{n};h_{n})+\mathbf{\xi }(t_{n},\mathbf{z}_{n};h_{n}),\quad with \quad
\mathbf{z}_{0}=\mathbf{x}_{0},</math>
where
<math>\mathbf{\phi }_{\mathbb{\gamma }}(t_{n},\mathbf{z}_{n};\delta
)=\int_{0}^{\delta }e^{\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n})(\delta
-u)}(\mathbf{f(}t_{n},\mathbf{z}_{n})+\mathbf{a}^{\mathbb{\gamma }}(t_{n},
\mathbf{z}_{n})u)du </math>
and
<math>\mathbf{\xi }\left( t_{n},\mathbf{z}_{n};\delta \right)
=\sum\limits_{i=1}^{m}\int\nolimits_{t_{n}}^{t_{n}+\delta }e^{\mathbf{f}_{
\mathbf{x}}(t_{n},\mathbf{z}_{n})(t_{n}+\delta -u)}\mathbf{g}_{i}(u)d\mathbf{
w}^{i}(u). </math>
Here,
<math>\mathbf{a}^{\mathbb{\gamma }}(t_{n},\mathbf{z}_{n})=
\left\{
\begin{matrix}
\mathbf{f}_{t}(t_{n},\mathbf{z}_{n}) & \qquad for \qquad \mathbb{\gamma }=1 \\
\mathbf{f}_{t}(t_{n},\mathbf{z}_{n})\quad +\frac{1}{2}\sum\limits_{j=1}^{m}
\left( \mathbf{I}_{d\times d}\times \mathbf{g}_{j}^{\intercal }
\left( t_{n}\right) \right) \mathbf{f}_{\mathbf{xx}}(t_{n},
\mathbf{z}_{n})\mathbf{g}_{j}\left( t_{n}\right) & \quad for \quad \mathbb{\gamma }=1.5,
\end{matrix}
\right.</math>
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