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t_{n}\right) \widetilde{J}_{\left( j,0\right) }
+\sum\limits_{j=1}^{m}\frac{d\mathbf{g}_{_{j}}}{dt}\left( t_{n}\right)
\widetilde{J}_{\left( 0,j\right) }</math></small>
<math>\qquad \qquad +\sum\limits_{j_{1},j_{2}=1}^{m}\left( \mathbf{I}_{d\times d}\otimes \mathbf{g}_{j_{2}}^{\intercal }\left(
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>
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>.
==== Order 2 SLL-RK schemes ====
For SDEs with a single Wiener noise '''(m=1)'''
<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) +
\mathbf{g}\left( t_{n}\right) \Delta w_{n}+\frac{\left( \mathbf{g}\left(
t_{n+1}\right) -\mathbf{g}\left( t_{n}\right) \right) }{h_{n}}J_{\left(
0,1\right) }. \qquad \qquad (18)</math>
where
<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}
_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y
}_{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>
<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}
_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y
}_{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>
with
<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>
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>.
==== Stability and dynamics ====
By construction the strong LL and HOLL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear ODEs, but it is not the case of the strong LL schemes in general. LL schemes '''(15)-(18)''' with <math>p\leq q\leq p+2 </math> are A-stable, which includes stiff and highly oscillatory linear equations. Moreover, for linear SDEs with [[Pullback attractor|random attractors]], these schemes also have a random attractor that [[Convergece of Random variables#Convergence in probability|converges in probability]] to the exact one as stepsizes decrease and preserve the [[ergodicity]] of these equations for any stepsize. These schemes also reproduce essential dynamical properties of simple and coupled harmonic oscillators such as the linear growth of energy along the paths, the oscillatory behavior around 0, the symplectic structure of Hamiltonian oscillators, and the mean of the paths. For nonlinear SDEs with small noise (e.g., '''(14)''' with <math>\mathbf{g}_{i}(t)\approx 0 </math>), the paths of these SLL schemes are basically the nonrandom paths of the LL scheme '''(6)''' for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the paths of the SLL scheme.
== Weak LL methods for SDEs ==
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