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Two approaches:PDE methodAbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupThe weak approximation problemSimulation – MC and QMCTwo important remarksOur Higher-order schemeFree Lie Algebra∂u(t, x) = Lu,∂twhere L = V 0 + (1/2) ∑ di=1 V 2i.Probabilistic method — “Simulation”u(0, x) = f(x).Step 1. Discretize X(t, x) and obtain X n (t, x).n = ♯{partitions of [0, T]}Euler–Maruyama, Higher order (Milstein, Kusuoka,..)Step 2. Integrate f(X n (T, x)) over D(n)-dimensional domain[0, 1) D(n) by MC, QMC, etc.D(n) depends on the discretization schemeWe consider only the probabilistic method here.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 AlgebraStep. 2 of the simulation is the numerical integration:E [ f ( X n (T, x) )] ∫= F(a 1 ,...,a D(n) ) da 1 ··· da D(n)[0,1) D(n)Calc. RHS by using MC or QMC.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation
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