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AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsOrder m Integration scheme: IS(m)⎧⎪⎨⎪⎩g ∈IS(m) ⇐⇒defg : C ∞ b (RN ; R N ) −→ ( R N → R N) ,∃ C m > 0 s.t. ∀ W ∈ C ∞ b (RN ; R N )sup |g(W)(x) − exp (W)(x)| ≤C m (‖W‖ C m+1) m+1x∈R N◮ IS(m) “Set of order m ODE solver”◮ We need to integrate, for example:Not necessarily s-linear.sV 0 +√ s2 (V 1 + ···+ V d ).S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation