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AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupThe weak approximation problemSimulation – MC and QMCTwo important remarksOur Higher-order schemeFree Lie AlgebraNon-commutative algebra (2)⎧⎪⎨∑⎫R〈〈A〉〉 :=a ⎪⎩ w w∣ a ⎪⎬w ∈ R⎪⎭ : R-algebra of formal series with basis A ∗ ,w∈A⎧∗ ⎪⎨∑⎫R〈A〉 := a ⎪⎩ w w ∈ R〈〈A〉〉∃ ⎪⎬k ∈ N s.t. a w = 0if|w| ≥k∣⎪⎭w∈A ∗:freeR-algebra with basis A ∗ ,∑(P, w) :=a w , for P = a w w ∈ R〈〈A〉〉,w∈A∑∗(PQ, w) := (P, u)(Q, v), for P, Q ∈ R〈〈A〉〉, w ∈ A ∗ .uv=wu,v∈A ∗S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation
AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupThe weak approximation problemSimulation – MC and QMCTwo important remarksOur Higher-order schemeFree Lie AlgebraNon-commutative algebra (3)◮ Projection, Truncation:∑j m (P) := (P, w)w‖w‖≤m∑P| k := (P, w)w|w|=k◮ Homogeneous component:R〈A〉 m := {P ∈ R〈A〉| (P, w) = 0if‖w‖ m}S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation
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