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Consider the ''d''-dimensional stochastic differential equation
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<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),\qquad t\in \left[ t_{0},T\right] , \qquad \qquad (19)</math>
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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.
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For a time discretization <math>\left( t\right) _{h}</math>, the order-<math>\mathbb{\beta }</math> <math>(=1,2)</math> ''Weak Local Linear discretization'' of the solution of the SDE '''(19)''' is defined by the recursive relation
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<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }_{\mathbb{\beta }}(t_{n},
\mathbf{z}_{n};h_{n})+\mathbf{\eta }(t_{n},\mathbf{z}_{n};h_{n}),\quad with \quad \mathbf{z}_{0}=\mathbf{x}_{0}, </math>
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where
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<math>\mathbf{\phi }_{\mathbb{\beta }}(t_{n},\mathbf{z}_{n};\delta
)=\int_{0}^{\delta }e^{\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})(\delta
-u)}(\mathbf{f(}t_{n},\mathbf{z}_{n})+\mathbf{b}^{\mathbb{\beta }}(t_{n},
\mathbf{z}_{n})u)du </math>
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with
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<math>\mathbf{b}^{\mathbb{\beta }}(t_{n},\mathbf{z}_{n})=
\begin{cases}{ll}
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.
</math>
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and <math>\mathbf{\eta }(t_{n},\mathbf{z}_{n};\delta )</math> is a zero mean stochastic process with variance matrix
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<math>\mathbf{\Sigma }(t_{n},\mathbf{z}_{n};\delta )=\int\limits_{0}^{\delta }e^{
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})(\delta -s)}\mathbf{G}(t_{n}+s)
\mathbf{G}^{\intercal }(t_{n}+s)e^{\mathbf{f}_{\mathbf{x}}^{\intercal
}(t_{n},\mathbf{z}_{n})(\delta -s)}ds. </math>
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Here, <math>\mathbf{f}_{\mathbf{x}}</math>, <math>\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, <math>\mathbf{f}_{\mathbf{xx}}</math> the Hessian matrix of <math>\mathbf{f}</math> with respect to <math>\mathbf{x}</math>, and <math>\mathbf{G}(t)=[\mathbf{g}_{1}(t),..., \mathbf{g}_{m}(t)]</math>. The weak Local Linear discretization <math>\mathbf{z}_{n+1}</math> [[Convergence of random variables|converges]] with order <math>\mathbb{\beta }</math> (=1,2) to the solution of '''(19)'''.
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==== Order 1 WLL scheme ====
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<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{B}_{14}+(\mathbf{B}_{12}\mathbf{B}
_{11}^{\intercal })^{1/2}\mathbf{\xi }_{n} </math>
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where, for SDEs with autonomous diffusion coefficients, <math>\mathbf{B}_{11}</math>, <math>\mathbf{B}_{12}</math> and <math>\mathbf{B}_{14}</math> are the submatrices defined by the [[Block matrix|partitioned matrix]] <math>\mathbf{B}=\mathbf{P}_{p,q}(2^{-k_{n}}\mathcal{M}_{n}h_{n}))^{2^{k_{n}}}</math>, with
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<math>\mathcal{M}_{n}=\left[
\begin{array}{cccc}
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\end{array}%
\right] \in \mathbb{R}^{(2d+2)\times (2d+2)}, </math>
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and <math>\{\mathbf{\xi }_{n}\}</math> is a sequence of ''d''-dimensional independent [[Bernoulli distribution|two-points distributed random vectors]] satisfying
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<math>P(\xi _{n}^{k}=\pm 1)=\frac{1}{2}. </math>
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==== Order 2 WLL scheme ====
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<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{B}_{16}+(\mathbf{B}_{14}\mathbf{B}
_{11}^{\intercal })^{1/2}\mathbf{\xi }_{n} </math>
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where <math>\mathbf{B}_{11}</math>, <math>\mathbf{B}_{14}</math> and <math>\mathbf{B}_{16}</math> are the submatrices defined by the partitioned matrix <math>\mathbf{B}=\mathbf{P} _{p,q}(2^{-k_{n}}\mathcal{M}_{n}h_{n}))^{2^{k_{n}}}</math> with
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<math>\mathcal{M}_{n}=\left[
\begin{array}{cccccc}
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\end{array}
\right] \in \mathbb{R}^{(4d+2)\times (4d+2)}, </math>
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<math>\mathbf{J}=\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\qquad
\mathbf{a}_{1}=\mathbf{f}(t_{n},\mathbf{y}_{n})\qquad \mathbf{a}
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\mathbf{I}\otimes (\mathbf{g}^{i}(t_{n}))^{\intercal })\mathbf{f}_{\mathbf{xx
}}(t_{n},\mathbf{y}_{n})\mathbf{g}^{i}(t_{n}) </math>
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and
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<math>\mathbf{H}_{0}=\mathbf{G}(t_{n})\mathbf{G}^{\intercal }(t_{n})\qquad
\mathbf{H}_{1}=\mathbf{G}(t_{n})\frac{d\mathbf{G}^{\intercal }(t_{n})}{dt}
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\mathbf{H}_{2}=\frac{d\mathbf{G}(t_{n})}{dt}\frac{d\mathbf{G}^{\intercal
}(t_{n})}{dt}\text{.} </math>
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==== Stability and dynamics ====
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