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Consider the ''d''-dimensional [[Stochastic differential equation|Stochastic Differential Equation]] (SDE)
<div style="text-align: center;">
<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>
</div>
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}
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the SDE (14) is defined by the recursive relation.
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<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>
</div>
where
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<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>
</div>
and
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<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>
</div>
Here,
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<math>\mathbf{a}^{\mathbb{\gamma }}(t_{n},\mathbf{z}_{n})=
\left\{
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\end{matrix}
\right.</math>
</div>
<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> [[Convergence of random variables|converges]] with order<math>\mathbb{\gamma } \quad (=1,1.5)</math> to the solution of '''(14)'''
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\mathbf{z}_{n})</math>, the equation for the residual <math>\mathbf{r}</math> is given by
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<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>
</div>
for all <math>t\in \lbrack t_{n},t_{n+1}]</math>, where
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<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}
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\mathbf{a}^{\gamma }\left( t_{n},\mathbf{z}_{n}\right) (s-t_{n})-\mathbf{f}
\left( t_{n},\mathbf{z}_{n}\right) . </math></small>
</div>
''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.
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<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>
</div>
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)'''.
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==== Order 1 SLL schemes ====
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<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+}\sum\limits_{i=1}^{m}\mathbf{g}
_{i}(t_{n})\Delta \mathbf{w}_{n}^{i}, (15)</math>
</div>
where the matrices <math>\mathbf{M}_{n}</math>, <math>\mathbf{L}</math> and <math>\mathbf{r}</math> are defined as in '''(6)''', <math>\Delta \mathbf{w}_{n}^{i}</math> is a [[Independent and identically distributed random variables|i.i.d.]] zero mean [[Normal distribution|Gaussian random variable]] with variance <math>h_{n}</math>, and '''p+q>1'''. For large systems of SDEs, in the above scheme <math>(\mathbf{P}_{p,q}(2^{-k_{n}}
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==== Order 1.5 SLL schemes ====
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<small><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}
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_{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>
</div>
where the matrices <math>\mathbf{M}_{n}</math>, <math>\mathbf{L}</math> and <math>\mathbf{r}</math> are defined as
<div style="text-align: center;">
<small><math>\mathbf{M}_{n}=
\begin{bmatrix}
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0 & 0 & 0
\end{bmatrix}
</div>
<div style="text-align: center;">
<math>\mathbf{L}=\left[
\begin{array}{ll}
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\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>.
</div>
==== Order 2 SLL-Taylor schemes ====
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<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}
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+\sum\limits_{j=1}^{m}\frac{d\mathbf{g}_{_{j}}}{dt}\left( t_{n}\right)
\widetilde{J}_{\left( 0,j\right) }</math></small>
</div>
<div style="text-align: center;">
<math>\qquad \qquad
+\sum\limits_{j_{1},j_{2}=1}^{m}\left(
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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> (17)
</div>
where <math>\mathbf{M}_{n}</math>, <math>\mathbf{L}</math>, <math>\mathbf{r}</math> and <math>\Delta \mathbf{w}_{n}^{i}</math> are defined as in the order-1 SLL schemes, and <math>\widetilde{J}_{\alpha } </math> is order-2 approximation to the multiple [[Stratonovish integral]] <math>J_{\alpha }</math>.
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For SDEs with a single Wiener noise '''(m=1)'''
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<math>\mathbf{y}_{t_{n+1}}=\mathbf{y}_{n}+\widetilde{\mathbf{\phi }}(t_{n},\mathbf{
y}_{n};h_{n})+\frac{h_{n}}{2}\left( \mathbf{k}_{1}+\mathbf{k}_{2}\right) +
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t_{n+1}\right) -\mathbf{g}\left( t_{n}\right) \right) }{h_{n}}J_{\left(
0,1\right) }. \qquad \qquad (18)</math>
</div>
where
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<math>\mathbf{k}_{1} =\mathbf{f}(t_{n}+\frac{h_{n}}{2},\mathbf{y}_{n}+\widetilde{
\mathbf{\phi }}(t_{n},\mathbf{y}_{n};\frac{h_{n}}{2})+\gamma _{+})-\mathbf{f}
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}_{n};\frac{h_{n}}{2})-\mathbf{f}\left( t_{n},\mathbf{y}_{n}\right) -\mathbf{
f}_{t}\left( t_{n},\mathbf{y}_{n}\right) \frac{h_{n}}{2},
</math>
</div>
<div style="text-align: center;">
<math>\mathbf{k}_{2} =\mathbf{f}(t_{n}+\frac{h_{n}}{2},\mathbf{y}_{n}+\widetilde{
\mathbf{\phi }}(t_{n},\mathbf{y}_{n};\frac{h_{n}}{2})+\gamma _{-})-\mathbf{f}
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}_{n};\frac{h_{n}}{2})-\mathbf{f}\left( t_{n},\mathbf{y}_{n}\right) -\mathbf{
f}_{t}\left( t_{n},\mathbf{y}_{n}\right) \frac{h_{n}}{2},</math>
</div>
with
<div style="text-align: center;">
<math>\gamma _{\pm }=\frac{1}{h_{n}}\mathbf{g}\left( t_{n}\right) \left\{
\widetilde{J}_{\left( 1,0\right) }\pm \sqrt{2\widetilde{J}_{\left(
1,1,0\right) }h_{n}-\widetilde{J}_{\left( 1,0\right) }^{2}}\right\} .
</math>
</div>
Here, <math>\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y}_{n};h_{n})=\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r} </math> for for low dimensional SDEs, and <math>\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y}_{n};h_{n})=\mathbf{L\mathbf{k}}_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}, \mathbf{r}) </math> for large systems of SDEs, where <math>\mathbf{M}_{n} </math>, <math>\mathbf{L} </math>, <math>\mathbf{r} </math>, <math>\Delta \mathbf{w}_{n}^{i} </math> and <math>\widetilde{J}_{\alpha } </math> are defined as in the order-2 SLL-Taylor schemes, '''p+q>1''' and <math>m_{n}>2 </math>.
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