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t_{0},t_{n}]</math> and by <math>\mathbf{y}\left( t\right) =\mathbf{\varphi }\left(
t\right)</math> for <math>t\in \left[ t_{0}-\tau ,t_{0}\right]</math>
For large systems of DDEs
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L\mathbf{k}}
_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}_{n},\mathbf{r})\quad and \quad \mathbf{y}\left( t\right) =\mathbf{y}_{n_{t}}+\mathbf{L\mathbf{k}}
_{m_{n_{t}},k_{n_{t}}}^{p,q}(t-t_{n_{t}},\mathbf{M}_{n_{t}},\mathbf{r}),</math>
with <math>p+q>1 \quad and \quad m_{n}>2.</math>
== LL methods for RDEs ==
Consider the ''d-dimensional'' Random Differential Equation (RDE)
<math>\frac{d\mathbf{x}\left( t\right) }{dt}=\mathbf{f}(\mathbf{x}(t),\mathbf{\xi }
(t)),\quad t\in \left[ t_{0},T\right] ,\qquad \qquad \qquad (13).
</math>
with initial condition <math>\mathbf{x}(t_{0})=\mathbf{x}_{0},</math> where <math>\mathbf{
\xi }</math> is a ''k''-dimensional separable finite continuous stochastic process, and
'''f''' is a differentiable function. Suppose that a realization (path) of <math>\mathbf{\xi }</math> is given.
=== Local Linear discretization ===
For a time discretization <math>\left( t\right) _{h}</math>, the ''Local Linear discretization'' of the RDE (13) at each point <math>t_{n+1}\in \left(
t\right) _{h}</math> is defined by the recursive expression
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}_{n};h_{n}),
\qquad with \qquad \mathbf{z}_{0}=\mathbf{x}_{0},</math>
where
<math>\mathbf{\phi }(t_{n},\mathbf{z}_{n};h_{n})=\int\limits_{0}^{h_{n}}e^{\mathbf{
f}_{\mathbf{x}}\left( \mathbf{z}_{n},\mathbf{\xi }(t_{n})\right) (h_{n}-u)}(
\mathbf{f(z}_{n},\mathbf{\xi }(t_{n}))+\mathbf{f}_{\mathbf{\xi }}(\mathbf{z}
_{n},\mathbf{\xi }(t_{n}))(\widetilde{\mathbf{\xi }}(t_{n}+u)-\widetilde{
\mathbf{\xi }}(t_{n})))du </math>
and <math>\widetilde{\mathbf{\xi }}</math> is an approximation to the process <math>\mathbf{
\xi }</math> for all <math>t\in \left[ t_{0},T\right]. </math> Here, <math>\mathbf{f}_{x}</math> and <math>\mathbf{f}_{\xi }</math> denote the partial derivatives with respect to <math>\mathbf{x}</math> and <math>\xi </math>, respectively.
=== Local Linearization schemes ===
Depending of the approximations <math>\widetilde{\mathbf{\xi }}</math> to the process <math>\mathbf{\xi }</math> and of the algorithm to compute <math>\mathbf{\phi }</math> different Local Linearizations schemes can be defined. Every numerical implementation <math>\mathbf{y}_{n}</math> of a Local Linear discretization <math>\mathbf{z}_{n}</math> is generically called ''Local Linearization scheme.''
==== LL schemes ====
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}
\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r,} </math> where the matrices <math>\mathbf{M}_{n}, \quad \mathbf{L} \quad and \quad \mathbf{r}</math> are defined as
<math>\mathbf{M}_{n}=\left[
\begin{array}{ccc}
\mathbf{f}_{\mathbf{x}}\left( \mathbf{y}_{n},\mathbf{\xi }(t_{n})\right) &
\mathbf{f}_{\mathbf{\xi }}(\mathbf{y}_{n},\mathbf{\xi }(t_{n})(\mathbf{\xi }
(t_{n+1})-\mathbf{\xi }(t_{n}))/h_{n} & \mathbf{f}\left( \mathbf{y}_{n},
\mathbf{\xi }(t_{n})\right) \\
0 & 0 & 1 \\
0 & 0 & 0%
\end{array}%
\right] \in \mathbb{R}^{(d+2)\times (d+2)},
</math>
<math>\mathbf{L}=\left[
\begin{array}{ll}
\mathbf{I}_{d} & \mathbf{0}_{d\times 2}
\end{array}
\right] </math>, <math>\mathbf{r}^{\intercal }=\left[
\begin{array}{ll}
\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}}
_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}_{n},\mathbf{r}),\quad p+q>1
\quad and \quad m_{n}>2.</math>
The convergence rate of both schemes is <math>min\{2,2\gamma \}</math>, where is <math>\gamma</math> the exponent of the Holder condition of <math>\mathbf{\xi }</math>.
== Strong LL methods for SDEs ==
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