Slides

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupOn an extension of an algorithm of higher-orderweak approximation to SDEsSyoiti Ninomiya 1 Mariko Ninomiya 21 Tokyo Institute of Technology2 The University of TokyoCREST and Sakigake International Symposium AsymptoticStatistics, Risk and Computation in Finance and Insurance2010 @Tokyo Institute of Technology (2010/12/14—18)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupOutlineAbstractBackgroundThe weak approximation problemSimulation – MC and QMCTwo important remarksOur Higher-order schemeFree Lie AlgebraMain resultL R ((A))-valued random variablesApproximation theoremThe AlgorithmsThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupS. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupBackground:◮ Anewhigher-orderweakapproximationschemebasedon[Kusuoka ’01] and [Lyons and Victoir ’04]Objective:◮ Construction of Concrete higher-order weak approximationalgorithms that are:◮◮Versatile, (applicable to a broad class of SDEs)Easy to use, (Blackbox algorithm)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupCurrent status:◮ Two kinds of schemes of order 2 (say Alg 1 [Victoir &N ’08] andAlg 2 [Ninomiya &N ’09]) Both work in practice.◮ General extrapolation method for Alg 1 [Oshima, Teichmann andVeluscek ’09]Enables arbitraly order weak approximationThis talk is on:◮ Construction of concrete Alg 2 algorithm of order > 2.◮ The state is under construction.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupReferences 1/2:◮ Kusuoka, S.: “Approximation of Expectation of Diffusion Processand Mathematical Finance.” In: T. Sunada (ed.) Advanced Studiesin Pure Mathematics, Proceedings of Final Taniguchi Symposium,Nara 1998, vol. 31, pp. 147–165 (2001)◮ Kusuoka, S.: “Approximation of Expectation of Diffusion Processesbased on Lie Algebra and Malliavin Calculs.” Advances inMathematical Economics 6, 69–83(2004)◮ Lyons, T., Victoir, N.: “Cubature on Wiener Space.” Proceedings ofthe Royal Society of London. Series A. Mathematical and PhysicalSciences 460, 169–198(2004)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupReferences 2/2:◮ N. and Victoir, N.: “Weak Approximation of Stochastic DifferentialEquations and Application to Derivative Pricing.” AppliedMathematical Finance 15, 107–121(2008)◮ Ninomiya, M. and N.: “A new weak approximation scheme ofstochastic differential equations by using the Runge–Kuttamethod”Finance and Stochastics 13, 415–443(2009)◮ Oshima, K., Teichmann, J. and Veluscek, D.: “A new extrapolationmethod for weak approximation schemes with applications”Preprint: arXiv:0911.4380v1 [math.PR] (2009)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 AlgebraThe Problem :Numerical calculation of E[f(X(T, x))], whereX(t, x) = x +d∑j=0∫ t0V j ∈ C ∞ b (RN ; R N )V j (X(s, x)) ◦ dB j (s)B(t) = (B 0 (t), B 1 (t),...,B d (t)),B 0 (t) = t, (B 1 (t),...,B d (t)) : d-dim Std. BM,(1)◦ dB j (s) :Stratonovichintegral.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

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

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupThe weak approximation problemSimulation – MC and QMCTwo important remarksOur Higher-order schemeFree Lie AlgebraMC and QMCW : r. v./(Ω, F , P), M ∈ Z >0MC(W, M) := 1 M∑W k , where {W i } M i=1M:iids.t.W 1 ∼ WQMC(W, M) := 1 Mk =1M∑W(ω k ), {ω i } ∞ i=1 :deterministicsequence(LDS)k =1MC(W, M) : r. v.QMC(W, M) ∈ R.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 AlgebraTwo types of approximation errors in simulation:1. Discritization error∣∣E [f(X(T, x))] − E [ f ( X n (T, x) )] ∣ ∣∣2. Integration error∣∣MC ( f ( X n (T, x) ) , M ) (ω) − E [ f ( X n (T, x) )] ∣ ∣∣or∣∣QMC ( f ( X n (T, x) ) , M ) − E [ f ( X n (T, x) )] ∣ ∣∣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 AlgebraIntegration error and MCBy CLT,MC ( f ( X n (T, x) ) , M ) ∼ NWhen we proceed simulation(E [ f ( X n (T, x) )] , Var [ f (X n (T, x)) ]Var [f(X(T, x))] ≈ Var [ f(X n (T, x)) ]Remark 1As long as we use MC, thenumberofsamplepointsM needed to attainthe given accuracy is independent of n and discretizing algorithm(Euler–Maruyama or Kusuoka etc.).M).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 AlgebraIntegration error and QMCThere exist sequences which satisfy∃ C f,n > 0 ∀ M ∈ Z >0∣∣QMC ( f ( X n (T, x) ) , M ) − E [ f ( X n (T, x) )] ∣ ∣∣ (log M) D(n)≤ Cf,D(n) .MRemark 2In contrast to the MC case, the number of sample points M needed bythe QMC to attain the given accuracy depends heavily on the dimensionof integration D(n). Smaller the dimension, smaller number of samplesare needed.⎧n × d Euler–Maruyama,⎪⎨D(n) = n × (d + 1) N. & Victoir,⎪⎩ 2n × d Ninomiya, & N.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 AlgebraOrder 1: Euler–Maruyama schemeX (EM),n (0, x) = x,X (EM),n ( k + 1n , x )= X (EM),n ( kn , x )+√Tnd∑i=0Ṽ i(X (EM),n ( kn , x ))Z i k +1 ,where,∀ jZ kj=⎧ √⎪⎨ T/n, if k = 0,⎪⎩ iid. r. v. ∼ N(0, 1), if k ∈{1,...,d},⎧⎪Ṽ i k (y) = ⎨⎪ ⎩V i (y) 0 + 1 2∑j V j V i (y) if k = 0,jif k ∈{1,...,d}.V i kS. 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 AlgebraApprox. Error of Euler–Maruyama scheme∣∣E [ f ( X(T, x) )] − [ ( E f X (EM),n (T, x) )] ∣ ∣∣∣= ( O n −1)= O (∆t)when f : bdd.& measurable and {V i } d : Unif. Hörmander Cond. [Bally & Talay ’96][Kohatsu-Higa ’00]i=0Euler–Maruyama scheme is an order 1 scheme.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 AlgebraIntuitive explanation of the scheme (1/3)( ) PXt f (x) := E [f (X(t, x))] , f ∈ C ∞ b (RN )L := V 0 + 1 d∑V 2j.2j=1S. 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 AlgebraIntuitive explanation of the scheme (2/3)Applying Ito formula repeatedly, we obtainE [f (X(t, x))] = ( P X t f ) (x) = f(x) +.= f(x) +=∫ t0∫ t0( ) PXs1Lf (x) ds 1{ ∫ s1 ((Lf)(x) +) }PXs2L 2 f (x) ds 2 ds 10n∑ ( ) (tL)if (x) + 1 i! n!i=0∫ t0(t − s) n ( P X s L n+1 f ) (x) ds.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 AlgebraIntuitive explanation of the scheme(3/3)n∑ ( ) (tL)iObservation: f (x) gives a nth order approx. of E [f(X(t, x))].i!i=0Slogan: Construct a random variable Ξ s. t.E[Ξ] =n∑ ( ) (tL)if (x).i!i=0S. 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 (1)A := {v 0 ,...,v d } :alphabetA ∗ := ( ∪ ∞ k =1 A k ) ∪{1} :freemonoidonAFor w = w 1 ···w k ∈ A ∗ , (w i ∈ A)|w| := k, ‖w‖ := |w| + card ({1 ≤ i ≤|w| ; w i = v 0 }) ,A ∗ m := {w ∈ A ∗ ||w| = m } ,A ∗ ≤m := {w ∈ A ∗ ||w| ≤m } .Concatenation product:For u = u 1 ···u k , v = v 1 ···v l ∈ A ∗ ,uv := u 1 ···u k v 1 ···v l .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 (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

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupThe weak approximation problemSimulation – MC and QMCTwo important remarksOur Higher-order schemeFree Lie AlgebraFree Lie algebra:[P, Q] := PQ − QP for P, Q ∈ R〈〈A〉〉,{}J A := K ⊂ R〈A〉sub R-module∣ A ⊂ K, [x, y] ∈ K ∀ x, y ∈ K,⋂L R (A) := : R-coefficients free Lie algebra on AK∈J AKL R ((A)) := { P ∈ R〈〈A〉〉 s.t. P| k ∈L R (A), ∀ k ∈ N } .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 AlgebraLogarithm and exponential:For P ∈ R〈〈A〉〉 s.t. (P, 1) = 0For Q ∈ R〈〈A〉〉 s.t. (Q, 1) = 1exp(P) := 1 +log(Q) :=∞∑k =1P kk! .∞∑ (−1) k −1 (Q − 1) k.kk =1log(exp(P)) = P and exp(log(Q)) = Q.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 AlgebraHausdorff product:For z 1 , z 2 ∈L R ((A)),z 1 ⊢⊣ z 2 := log(exp(z 1 ) exp(z 2 )).◮ (z 1 ⊢⊣ z 2 ) ⊢⊣ z 3 = log (exp(z 1 ) exp(z 2 ) exp(z 3 )) = z 1 ⊢⊣ (z 2 ⊢⊣ z 3 )=: z 1 ⊢⊣ z 2 ⊢⊣ z 3◮ z 2 , z 1 ∈L R =⇒ z 2 ⊢⊣ z 1 ∈L R ((A))∵ the Baker–Campbell–Hausdorff–Dynkin formula.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 AlgebraL R (A) and R〈A〉Element of L R ⇐⇒ Vector field generated by V 0 ,...,V dElements of R〈A〉 ⇐⇒Differential operator generated by V 0 ,...,V dS. 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 AlgebraResurgence and rescaling◮ Resurgence operator Φ:Φ(1) := id,Φ(v i1 ···v in ):= V i1 ···V in◮ Rescaling operator: For s > 0, Ψ s : R〈〈A〉〉 → R〈〈A〉〉 is defined as:∞∑ ∞∑Ψ s⎛⎜⎝ P m⎞⎟⎠ = s m/2 P m where P m ∈ R〈A〉 m .m=0 m=0Example:(Φ(Ψ s v 0 + 1 ))2 [v 1, v 2 ] = sV 0 + s 2 [V 1, V 2 ]S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsNotation: exp(V)(x)For a smooth vector field V, (i.e.V ∈ C ∞ b (RN ; R N ))where y(t) is a solution to the ODE:exp(V)(x) := y(1)y(0) = x,dy(t)dt= V(y(t))S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

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

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsR〈〈A〉〉, L R ((A))-valued random variables◮ Topology: R〈〈A〉〉 ≈ R ∞ –Directproducttopology◮ R〈〈A〉〉-valued and L R ((A))-valued probability theory can beconsidered.(Ito formula, etc.)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsApproximation theorem (1/2)m ≥ 1, M ≥ 2,Z 1 ,...,Z M : L R ((A))-valued random variables s. t.Z i = j m Z i for i = 1,...,M,E [ ]‖j m Z i ‖ 2 < ∞ for i = 1,...,M,⎡M∑E ⎢⎣⎛⎜⎝ exp a ∥∥Φ ( ⎞⎤( ))∥ ∥∥∥CΨ s Zjm+1⎟⎠⎥⎦ < ∞ for any a > 0.j=1S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

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

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsCor. to the Approximation theorem“Q (s) gives (m − 1)/2-order weak approximation.”Let Q (s) for s ∈ (0, 1] be(Q(s) f ) (x) := E [ f (g (Φ (Ψ s (Z 1 ))) ◦···◦g (Φ (Ψ s (Z M ))) (x)) ] ,where f ∈ C ∞ b (RN ; R) and g ∈IS(m) then ∃ C > 0,∥∥P s f − ∥ Q (s) f ∥∞ ≤ Cs (m+1)/2 ‖grad(f)‖ ∞P s (f) := E [f(X(t, x))](Remark: When {V 0 ,...,V d } finitely generated.)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsAlgorithm 1 [Victoir–N.(2004)](Λ i , Z i ) i∈{1,··· ,n} :2n indep. r. v.,∀ iP(Λ i = ±1) = 1 2 , Z i ∼ N(0, I d ).{X (Alg.1),n }k k =0,...,n :afamilyofr.v.definedas:X (Alg.1),n0:= x,X (Alg.1),n :=(k +1)/n⎧ (V0) ⎛exp exp ⎜⎝ Z 1 k V ⎛1√⎞⎟⎠ ···exp ⎜⎝ Z d k V ( )d V0√⎞⎟⎠ exp⎪⎨ 2n n n 2n( ) ⎛V0⎪⎩ exp exp ⎜⎝ Z d k V ⎛d√⎞⎟⎠ ···exp ⎜⎝ Z 1 k V ( )1 V0√⎞⎟⎠ exp2n n n 2nX (Alg.1),nkif Λ k =+1,X (Alg.1),nkif Λ k = −1.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsExtrapolations of Algorithm 1◮ To an arbitrary order [Oshima, Teichmann and Veluscek ’09]◮ To the 6th order [Fujiwara ’06]S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsGeneral framework of Algorithm 2 (1)◮ Remind the Slogan◮ Find the Ξ written as:Ξ=Z 1 ⊢⊣ Z 2 ⊢⊣ · · ⊢⊣ · Z Md∑Z j = c j v 0 + S i j v i,i=1j ∈{1,...,M}wherec j ∈ R s.t. c 1 + ···+ c M = 1E [ S i j Si′ j ′ ]= Rjj ′δ ii ′S i j∼ N(0, R jj )(i ∈{1,...,d}, j ∈{1,...,M})S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsGeneral framework of Algorithm 2 (2)The slogan is equivalent to:E [j m (exp(Z 1 ) ···exp(Z M ))] = j m⎛⎜⎝ exp ⎛⎜⎝ v 0 + 1 2d∑i=1v 2 i⎞⎞⎟⎠ ⎟⎠ (2)The problem:Find real numbers {c j } { M j=1 , R ij that satisfy (2).}1≤i≤j≤M#unknown vars = 1 M(M + 3)2S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsMain tools (Theorem 1):For the LHS of (2)If n w (i) is odd for some i ∈{1,...,d}, thenC(w) = 0.If n w (i) is even for every i ∈{1,...,d}, thenC(w) =∑⃗k=(k 1 ,...,k M )∈K l (M)×⎛d∏p=1⎜⎝1k 1 ! ···k M !∑M∏ ( ) ( )N w0,j,⃗ kcjj=1{d ij} 1≤i≤j≤M∈e ( N w ( p,1,⃗ k),...,Nw ( p,M,⃗ k))∏ M2 − ∑ Mi=1 d j=1ii( ( ) )Nwp, j, ⃗ k ! ∏∏ (1≤i≤j≤M dij ! ) 1≤i≤j≤MR d ijij⎞⎟⎠(3)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsMain tools (Theorem 2):For the RHS of (2)Let A 0 = {v 0 , v 1 v 1 , v 2 v 2 ,...,v d v d }⊂A ∗ .Then⎛⎞exp ⎜⎝ v 0 + 1 d∑ ∑v 2 i2⎟⎠ =1i=1 w=w 1···w l2 |w|−l l! w,w 1 ,...,w l ∈A 0that is,⎧1 ⎪⎨ if w ∈ A 0C(w) =2⎪⎩|w|−l l!0 otherwise.(4)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsAlgebraic relations between {c j } { }Mj=1 , R ij1≤i≤j≤MUsing following 3 results, the algebraic relations are obtained.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsAn example of Algorithm 2 [Ninomiya–N. (2009)]When (m, M) = (5, 2),c 1 = ∓ √ √2 (2u − 1)2 (2u − 1), c 2 = 1 ±22R 22 = 1 + u ± √ √ 2 (2u − 1)2 (2u − 1), R 12 = −u ∓2Becomes 1-dimensional ideal.R 11 = u for some u ≥ 1/2.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsPartial results (1):The case m ≥ 6◮ (m, M) = (7, 2): idealbecomestrivial(i.e. = C [c 1 , c 2 , R 11 , R 12 , R 13 ])◮ (m, M, d) = (7, 3, 1): idealbecomestrivial.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsPartial results (2):The case (m, M, d) = (7, 3, 2) and v 0 free:The algebraic relation:14R 22 + 24R 13 − 13, 4R 33 + 14R 22 + 4R 11 − 15, 36R 33 2 − 60R 22 + 25,−72R 22 R 33 + 68R 33 + 24R 23 + 22R 22 − 17,−288R 23 R 33 − 8R 33 + 264R 23 − 46R 22 + 77,12R 23 − 22R 22 + 12R 12 + 23, 8R 33 + 216R 22 R 23 − 156R 23 + 34R 22 − 27,Unfortunately, the solution is:−22R 33 − 324R 23 2 − 192R 23 + 46R 22 − 57,144R 33 2 − 64R 33 + 168R 23 + 46R 22 + 7R 33 = 5 ± √ −1112 RS. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsPartial results (3):The case (m, M, d) = (7, 4, 3) and v 0 free:Over 1-month symbolical calculation cannot find the answer.....S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsWe have some more results (Theorem 3): A ∗ = ⋃ ∞i=0 A i ,⋔ : A ∗ ⊗ A ∗ → A ∗ :Shuffleproduct.(A ∗ , ⋔) becomes an algebra (shuffle algebra).C(·)is ring homomorphism.Cor.C : (A ∗ , ⋔) −→ R [ c j , R ij , (1 ≤ i ≤ j ≤ M) ]C ( (0) n 0⋔ (11) ⋔ ···⋔ (ll) ) = C(0) n 0C(11) lS. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsAn(example)of Algorithm 2 [Ninomiya–N. (2005)]j Z ,i,ki∈{1,...,d},j∈{1,2}k ∈{0,...,n−1}( ) ( √ )( )Z1i,k 1/2 1/ 2 ηZ 2 =i,k 1/2 −1/ √ 1i,k2 η 2 , where η j i.i.d.∼ N(0, 1).i,ki,k{X (Alg.2),n }k k =0,...,n :afamilyofr.v.definedas:X (Alg.2),n0:= x,X (Alg.2),n :=(k +1)/n⎛1exp ⎜⎝ 2n V 0 +d∑i=1Z 1i,k√ V in⎞⎟⎠⎛⎜⎝ exp 12n V 0 +d∑i=1Z Mi,k√ nV i⎞⎟⎠ X (Alg.2),nk /nS. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsTheoremBoth X (Alg.1),n and X (Alg.2),n are of order 2.Recent result by KusuokaIf Q (s) is constructed by X (Alg.1),n or X (Alg.2),n then ∃ C > 0s.t.(P s f) (x) − ( Q (s) f ) (x) = Cs (m+1)/2 + O ( s (m+3)/2)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupL R ((A))-valued random variablesApproximation theoremThe AlgorithmsD(n)D(n) :dimensionofintegrationdomain⎧n × d Euler–Maruyama,⎪⎨D(n) = n × (d + 1) Algorithm 1⎪⎩ n × 2 × d Algorithm 2S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupHow to calc. exp(Z i )(x)?◮ Lucky case: We can get exact form of exp(Z i )(x).Often the case with Algorithm 1.◮ Otherwise: Forced to proceed with nuerical approximation.Good news:Theorem [Ninomiya–N.]Classical order m Runge–Kutta methods belongs to IS(m)S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupMC or QMC with Algorithms 1 and 2◮ W: “asetofODEs”-valuedr.v.byAlgorithm1or2◮ MC or QMC:◮◮Draw a set of ODEs “W(ω)” fromW and obtain (somethinglike) exp(W(ω))(x) numericallyIterate the step above and calculate the average:1MM∑exp(W(ω i ))(x)i=1S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupAdvantages of the approache◮ Free from symbolical calculation.◮◮Calc. in group is easy.◮ Numerical ODE solver works (by the 2nd th’m)Calc. in algebra is difficult.◮◮Huge symbolical calc.Simultaneous distributions of multiple integrations of BMs arenot knowm. (except for the 2nd order.)◮ Universal (applicable to non-commutative {V i } d i=0 )Naïve “Ito-Taylor expansion with the Runge–Kutta” suffers from thenon-commutativity.S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupNumerical Example:Pricing of Asian option under the Heston model:Heston model:Y 1 (t, x) = x 1 +Y 2 (t, x) = x 2 +〈W 1 , W 2〉 = ρtt∫ t0∫ t0µY 1 (s, x) ds +∫ tα (θ − Y 2 (s, x)) ds +0Y 1 (s, x) √ Y 2 (s, x) dW 1 (s),∫ t0β √ Y 2 (s, x) dW 2 (s),Asian Option:Y 3 (t, x) =∫ t0Y 1 (s, x) ds,( )Y3 (T, x)Payoff = max − K, 0 .TS. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupDiscretization Error0.010.001Discretization Error and Num. of PartitionsEuler-MaruyamaEuler-Maruyama + RombergNinomiya-VictoirNinomiya-Victoir + RombergNew MethodNew Method + RombergO(1/n 2 )O(1/n 3 )5e-055e-061e-04Error1e-051e-061e-071 10 100 1000 10000 100000S. Ninomiya, Num. of M. Partitions Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupConvergfence Error from quasi-Monte Carlo and Monte Carlo2σ(for MC) and Absolute difference(for QMC)10.10.010.0011e-041e-05QMC : New Method + Romberg, 1/Δt=2QMC : Ninomiya-Victoir + Romberg, 1/Δt=4QMC : New Method, 1/Δt=10QMC : Ninomiya-Victoir, 1/Δt=12QMC :Euler-Maruyama + Romberg, 1/Δt=16MC : Euler-Maruyama5e-055e-061e-0610 100 1000 10000 100000 1e+06 1e+07S. Ninomiya, Num. of M. Sample Ninomiya points Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupOverall performance comparison:#Partition, #Dim, #Sample, and CPU time required for 10 −4 accuracy.Method #Partition #Dim #Sample CPU time (sec)E-M + MC 2000 4000 10 8 1.72 × 10 5E-M + Extrpltn + QMC 16 + 8 48 5× 10 6 1.27 × 10 2N-V + QMC 12 36 2 × 10 5 3.24N-V + Extrpltn + QMC 4 + 2 18 2× 10 5 1.76KNN + QMC 10 40 2 × 10 5 3.4KNN + Extrpltn + QMC 2 + 1 12 2× 10 5 1.2S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupRecent developments:◮ Existence of the asymptotic expansion of the errors of Algorithms 1and 2 [Kusuoka]◮ Order 6 algorithm [Fujiwara]◮ SPDE case [Teichmann]S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation

AbstractBackgroundMain resultThe Runge–Kutta methodNumerical ExampleRecent developmentsAppendix/BackupBoth approaches are necessary because:◮ PDE approach works only when◮◮L is elliptic,dimesion is small.◮ Simulation is the last resort but (people believes) very timeconsuming.This presentation focuses on “Simulation.”S. Ninomiya, M. Ninomiya Extension of a higher-order weak approximation