Slides

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsApproximation theorem (2/2)=⇒∀ p ∈ [1, ∞), ∀ g 1 ,..., ∀ g M ∈IS(m), ∃ C m,M > 0s.t.∥ sup |g 1 (Φ (Ψ s (Z 1 ))) ◦···◦g M (Φ (Ψ s (Z M ))) (x)x∈R N− exp (Φ (Ψ s (j m (Z M ⊢⊣ · · ⊢⊣ · Z 1 )))) (x) ∣ ∥ ∥∥∥∥L∣ ≤ C m,M s (m+1)/2pfor ∀ s ∈ (0, 1] where C m,M depends only on m and M.f ◦ g(x) := f (g(x))S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation