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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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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 ===
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\end{matrix}
\right.</math>
<math>\mathbf{f}_{\mathbf{x}}, \mathbf{f}_{t}</math> denote the partial derivatives of <math>\mathbf{f}</math> with respect to the variables <math>\mathbf{x}</math> and '''''t''''', respectively, and <math>\mathbf{f}_{\mathbf{xx}}</math> the Hessian matrix of <math>\mathbf{f}</math> with respect to <math>\mathbf{x}</math>. The strong Local Linear discretization <math>\mathbf{z}_{n+1}</math> converges with order<math>\mathbb{\gamma } \quad (=1,1.5)</math> to the solution of '''(14)'''
=== High Order Local Linear discretizations ===
After the local linearization of the drift term of '''(14)''' at <math>(t_{n},
\mathbf{z}_{n})</math>, the equation for the residual <math>\mathbf{r}</math> is given by
<math>d\mathbf{r}\left( t\right) =\mathbf{q}_{\gamma }(t_{n},\mathbf{z}_{n};t
\mathbf{,\mathbf{r}}\left( t\right) )dt+\sum\limits_{i=1}^{m}\mathbf{g}
_{i}(t)d\mathbf{w}^{i}(t)\mathbf{,}\qquad \mathbf{r}\left( t_{n}\right)
=\mathbf{0} </math>
for all <math>t\in \lbrack t_{n},t_{n+1}]</math>, where
<small><math>\mathbf{q}_{\gamma }(t_{n},\mathbf{z}_{n};s\mathbf{,\xi })=\mathbf{f}(s,
\mathbf{z}_{n}+\mathbf{\phi }_{\gamma }\left( t_{n},\mathbf{z}
_{n};s-t_{n}\right) +\mathbf{\xi })-\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}
_{n})\mathbf{\phi }_{\gamma }\left( t_{n},\mathbf{z}_{n};s-t_{n}\right) -
\mathbf{a}^{\gamma }\left( t_{n},\mathbf{z}_{n}\right) (s-t_{n})-\mathbf{f}
\left( t_{n},\mathbf{z}_{n}\right) . </math></small>
''High Order Local Linear discretization'' of the SDE '''''(14)''''' at each point <math>t_{n+1}\in \left( t\right) _{h} </math> is then defined by the
recursive expression.
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }_{\gamma }(t_{n},\mathbf{z}
_{n};h_{n})+\widetilde{\mathbf{r}}(t_{n},\mathbf{z}_{n};h_{n}),\qquad with \qquad
\mathbf{z}_{0}=\mathbf{x}_{0}, </math>
where <math>\widetilde{\mathbf{r}} </math> is strong approximation to the residual <math>\mathbf{r} </math> of order <math>\alpha </math> higher than '''1.5'''. The strong HOLL discretization <math>\mathbf{z}_{n+1} </math> converges with order <math>\alpha </math> to the solution of '''(14)'''.
=== Local Linearization schemes ===
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