Adapting quasi-monte Carlo methods to simulation problems in weighted Korobov spaces

Transcription

1 Adapting quasi-monte Carlo methods to simulation problems in weighted Korobov spaces Christian Irrgeher joint work with G. Leobacher RICAM Special Semester Workshop 1 Uniform distribution and quasi-monte Carlo methods October 2013, Linz Christian Irrgeher (JKU Linz) 1

2 Problem formulation Efficient computation of E(g(B)) B... standard Brownian motion with index set [0, T] g... suitable function Christian Irrgeher (JKU Linz) 2

3 Problem formulation Efficient computation of E(g(B)) B... standard Brownian motion with index set [0, T] g... suitable function Examples in finance, biology, physics,... e.g.: Financial derivative pricing Gaussian financial market models European-style options Christian Irrgeher (JKU Linz) 2

4 Numerical simulation quasi-monte Carlo (QMC) 1. Discretization E(g(B)) E(gd (B T d (X1,..., X d ) are independent N (0, 1),..., B d T )) = E(f d (X 1,..., X d )) =: I(f d ) d Christian Irrgeher (JKU Linz) 3

5 Numerical simulation quasi-monte Carlo (QMC) 1. Discretization E(g(B)) E(gd (B T d (X1,..., X d ) are independent N (0, 1) 2. QMC integration I(fd ) 1 N N j=1 f d(x j ) =: Q d,n (f d ) {x1,..., x N } R d deterministic point set,..., B d T )) = E(f d (X 1,..., X d )) =: I(f d ) d Christian Irrgeher (JKU Linz) 3

6 Numerical simulation quasi-monte Carlo (QMC) 1. Discretization E(g(B)) E(gd (B T d (X1,..., X d ) are independent N (0, 1) 2. QMC integration I(fd ) 1 N N j=1 f d(x j ) =: Q d,n (f d ) {x1,..., x N } R d deterministic point set,..., B d T )) = E(f d (X 1,..., X d )) =: I(f d ) d Error of QMC algorithm Q d,n err := E(g(B)) Q d,n (f d ) Christian Irrgeher (JKU Linz) 3

7 Error estimate First estimate: err E(g(B)) I ( ) f d + I ( ) f d Qd,N (f d ) discretization error integration error Christian Irrgeher (JKU Linz) 4

8 Error estimate First estimate: err E(g(B)) I ( ) f d + I ( ) f d Qd,N (f d ) discretization error integration error Analysis of both errors emphasis on integration error but discretization error not negligible Christian Irrgeher (JKU Linz) 4

9 Discretization error Discretization (with step size 1/d) Euler-Maruyama method Milstein method... Christian Irrgeher (JKU Linz) 5

10 Discretization error Discretization (with step size 1/d) Euler-Maruyama method Milstein method... Discretization error err disc c 1 d p with convergence rate p > 0 and constant c 1 > 0 Christian Irrgeher (JKU Linz) 5

11 Discretization error Discretization (with step size 1/d) Euler-Maruyama method Milstein method... Discretization error err disc c 1 d p with convergence rate p > 0 and constant c 1 > 0 Convergence rate depends on discretization method function g Christian Irrgeher (JKU Linz) 5

12 Gaussian measure and Hermite polynomials Density of the (standard) Gaussian measure ϕ(x) = 1 2π e x x 2 Christian Irrgeher (JKU Linz) 6

13 Gaussian measure and Hermite polynomials Density of the (standard) Gaussian measure ϕ(x) = 1 2π e x x 2 L 2 (R d, ϕ) = {f : R d R : f measurable, R d f(x) 2 ϕ(x)dx < } Christian Irrgeher (JKU Linz) 6

14 Gaussian measure and Hermite polynomials Density of the (standard) Gaussian measure ϕ(x) = 1 2π e x x 2 L 2 (R d, ϕ) = {f : R d R : f measurable, R d f(x) 2 ϕ(x)dx < } Univariate Hermite polynomials H k (x) = ( 1)k e x2 2 k! d k dx k e x 2 2 Christian Irrgeher (JKU Linz) 6

15 Gaussian measure and Hermite polynomials Density of the (standard) Gaussian measure ϕ(x) = 1 2π e x x 2 L 2 (R d, ϕ) = {f : R d R : f measurable, R d f(x) 2 ϕ(x)dx < } Univariate Hermite polynomials H k (x) = ( 1)k e x2 2 k! Multivariate Hermite polynomials d k dx k e x 2 2 d H k (x) = H kj (x j ) j=1 Christian Irrgeher (JKU Linz) 6

16 Gaussian measure and Hermite polynomials Density of the (standard) Gaussian measure ϕ(x) = 1 2π e x x 2 L 2 (R d, ϕ) = {f : R d R : f measurable, R d f(x) 2 ϕ(x)dx < } Univariate Hermite polynomials H k (x) = ( 1)k e x2 2 k! Multivariate Hermite polynomials {H k } k is an ONB of L 2 (R d, ϕ) d k dx k e x 2 2 d H k (x) = H kj (x j ) j=1 Christian Irrgeher (JKU Linz) 6

17 Hermite expansion Hermite expansion of f L 2 (R d, ϕ) f(x) = ˆf(k)H k (x) in L 2 k N d 0 k-th Hermite coefficient ˆf(k) = R d f(x)h k (x)ϕ(x)dx Christian Irrgeher (JKU Linz) 7

18 Hermite expansion Hermite expansion of f L 2 (R d, ϕ) f(x) = ˆf(k)H k (x) in L 2 k N d 0 k-th Hermite coefficient ˆf(k) = R d f(x)h k (x)ϕ(x)dx Theorem Let f L 2 (R d, ϕ) C(R d ) and k N ˆf(k) d <. Then 0 f(x) = ˆf(k)H k (x) for all x R d. k N d 0 Christian Irrgeher (JKU Linz) 7

19 Korobov space of functions on R Let α > 1, γ > 0. Define for k N 0 : { 1 if k = 0 r(α, γ, k) := γk α if k 0 Christian Irrgeher (JKU Linz) 8

20 Korobov space of functions on R Let α > 1, γ > 0. Define for k N 0 : r(α, γ, k) := Introduce inner product: { 1 if k = 0 γk α if k 0 f, g α,γ := r(α, γ, k) 1 ˆf(k)ĝ(k) k=0 Christian Irrgeher (JKU Linz) 8

21 Korobov space of functions on R Let α > 1, γ > 0. Define for k N 0 : r(α, γ, k) := { 1 if k = 0 γk α if k 0 Introduce inner product: f, g α,γ := r(α, γ, k) 1 ˆf(k)ĝ(k) k=0 Corresponding norm: f α,γ = f, f α,γ Christian Irrgeher (JKU Linz) 8

22 Korobov space of functions on R Let α > 1, γ > 0. Define for k N 0 : r(α, γ, k) := { 1 if k = 0 γk α if k 0 Introduce inner product: f, g α,γ := r(α, γ, k) 1 ˆf(k)ĝ(k) k=0 Corresponding norm: f α,γ = Function space: f, f α,γ H α,γ (R, ϕ) := {f L 2 (R, ϕ) C(R) : f α,γ < } Christian Irrgeher (JKU Linz) 8

23 Korobov space of functions on R H α,γ (R, ϕ) is a reproducing kernel Hilbert space Christian Irrgeher (JKU Linz) 9

24 Korobov space of functions on R H α,γ (R, ϕ) is a reproducing kernel Hilbert space Reproducing kernel function K α,γ : R R R Kα,γ (, y) H α,γ (R, ϕ) y R f, Kα,γ (, y) α,γ = f(y) y R f H α,γ (R, ϕ) Christian Irrgeher (JKU Linz) 9

25 Korobov space of functions on R H α,γ (R, ϕ) is a reproducing kernel Hilbert space Reproducing kernel function K α,γ : R R R Kα,γ (, y) H α,γ (R, ϕ) y R f, Kα,γ (, y) α,γ = f(y) y R f H α,γ (R, ϕ) Series representation of the reproducing kernel K α,γ (x, y) = 1 + γ k α H k (x)h k (y) k=1 Christian Irrgeher (JKU Linz) 9

26 Korobov space of functions on R There are interesting functions in this space. Define differential operator D x := d dx x Christian Irrgeher (JKU Linz) 10

27 Korobov space of functions on R There are interesting functions in this space. Define differential operator D x := d dx x Theorem (I. & Leobacher) Let β > 2 be an integer and f : R R be a β times differentiable function such that (i) D j xf(x)ϕ(x) 1 2 L 1 (R) for each j {1,..., β} and (ii) D j xf(x) = O ( e x2 /(2c) ) as x for each j {0,..., β 1} and some c > 1. Then f H α,γ (R, ϕ) with 1 < α < β 1. Christian Irrgeher (JKU Linz) 10

28 Korobov space of functions on R d For non-increasing weights γ = (γ 1,..., γ d ) H α,γ (R d, ϕ) := H α,γ1 (R, ϕ)... H α,γd (R, ϕ) Christian Irrgeher (JKU Linz) 11

29 Korobov space of functions on R d For non-increasing weights γ = (γ 1,..., γ d ) H α,γ (R d, ϕ) := H α,γ1 (R, ϕ)... H α,γd (R, ϕ) Inner product: f, g α,γ = k N d r(α, γ, k) 1 ˆf(k)ĝ(k) 0 with r(α, γ, k) = d j=1 r(α, γ j, k j ) Christian Irrgeher (JKU Linz) 11

30 Korobov space of functions on R d For non-increasing weights γ = (γ 1,..., γ d ) H α,γ (R d, ϕ) := H α,γ1 (R, ϕ)... H α,γd (R, ϕ) Inner product: f, g α,γ = k N d 0 r(α, γ, k) 1 ˆf(k)ĝ(k) with r(α, γ, k) = d j=1 r(α, γ j, k j ) H α,γ (R d, ϕ) is a weighted reproducing kernel Hilbert space Reproducing kernel Kα,γ (x, y) = d j=1 K α,γ j (x j, y j ) Influence of variables is given by the choice of weights γ Christian Irrgeher (JKU Linz) 11

31 Integration Error Let f d H α,γ (R d, ϕ). I(fd ) = f R d d (x)ϕ(x)dx = f R d d, K α,γ (, x) α,γ ϕ(x)dx = f d, K R d α,γ (, x)ϕ(x)dx α,γ Qd,N (f d ) = f d, 1 N N n=1 K α,γ(, x n ) α,γ Christian Irrgeher (JKU Linz) 12

32 Integration Error Let f d H α,γ (R d, ϕ). I(fd ) = R d f d (x)ϕ(x)dx = R d f d, K α,γ (, x) α,γ ϕ(x)dx = f d, R d K α,γ (, x)ϕ(x)dx α,γ Qd,N (f d ) = f d, 1 N N n=1 K α,γ(, x n ) α,γ Integration error err QMC = I(f d ) Q d,n (f d ) f α,γ d K α,γ (, x)ϕ(x)dx 1 N K α,γ (, x n ) R d N α,γ n=1 }{{} =:e d,n (x 1,...,x N ) Christian Irrgeher (JKU Linz) 12

33 Worst case error of integration Considering the Gaussian-weighted root-mean-square error for QMC integration: ( ē d,n := e 2 d,n(x 1,..., x N )ϕ(x 1 )... ϕ(x N )d(x 1,..., x N ) R dn ) 1 2 Christian Irrgeher (JKU Linz) 13

34 Worst case error of integration Considering the Gaussian-weighted root-mean-square error for QMC integration: ( ē d,n := e 2 d,n(x 1,..., x N )ϕ(x 1 )... ϕ(x N )d(x 1,..., x N ) R dn Theorem (I. & Leobacher) The Gaussian-weighted root-mean-square error for QMC integration in the Korobov space H α,γ (R d, ϕ) is ē d,n = 1 d (1 + γ j ζ(α)) 1 N j= ) 1 2 Christian Irrgeher (JKU Linz) 13

35 Worst case error of integration and tractability There exists a point set {x 1,..., x N } such that e d,n 1 ( ζ(α) exp N 2 d ) γ j j=1 Christian Irrgeher (JKU Linz) 14

36 Worst case error of integration and tractability There exists a point set {x 1,..., x N } such that e d,n 1 ( ζ(α) exp N 2 Strong tractability, if lim sup d dj=1 γ j <. d ) γ j j=1 Christian Irrgeher (JKU Linz) 14

37 Worst case error of integration and tractability There exists a point set {x 1,..., x N } such that e d,n 1 ( ζ(α) exp N 2 Strong tractability, if lim sup d dj=1 γ j <. d ) γ j j=1 Polynomial tractability, if lim sup d d j=1 γ j ln(d) <. Christian Irrgeher (JKU Linz) 14

38 Worst case error of integration and tractability There exists a point set {x 1,..., x N } such that e d,n 1 ( ζ(α) exp N 2 Strong tractability, if lim sup d dj=1 γ j <. d ) γ j j=1 Polynomial tractability, if lim sup d d j=1 γ j ln(d) <. err c 1 d p + f d α,γ c 2 N 1/2 d q f d α,γ can grow in d Christian Irrgeher (JKU Linz) 14

39 Orthogonal transforms For any orthogonal transform U on R d : f d (x)ϕ(x)dx = f d (Ux)ϕ(x)dx R d R d Christian Irrgeher (JKU Linz) 15

40 Orthogonal transforms For any orthogonal transform U on R d : f d (x)ϕ(x)dx = f d (Ux)ϕ(x)dx R d R d Every orthogonal transform corresponds to an Brownian path construction (Papageorgiou 2002) Christian Irrgeher (JKU Linz) 15

41 Orthogonal transforms For any orthogonal transform U on R d : f d (x)ϕ(x)dx = f d (Ux)ϕ(x)dx R d R d Every orthogonal transform corresponds to an Brownian path construction (Papageorgiou 2002) Classical construction methods Forward construction Brownian bridge construction PCA construction Christian Irrgeher (JKU Linz) 15

42 Orthogonal transforms For any orthogonal transform U on R d : f d (x)ϕ(x)dx = f d (Ux)ϕ(x)dx R d R d Every orthogonal transform corresponds to an Brownian path construction (Papageorgiou 2002) Classical construction methods Forward construction Brownian bridge construction PCA construction Equivalence principle (Wang & Sloan 2011) Roughly spoken: every construction that is good for one function is bad for another Christian Irrgeher (JKU Linz) 15

43 Orthogonal transforms Bound of the integration error: err QMC f d U α,γ e d,n Christian Irrgeher (JKU Linz) 16

44 Orthogonal transforms Bound of the integration error: err QMC f d U α,γ e d,n In general, f d α,γ f d U α,γ Christian Irrgeher (JKU Linz) 16

45 Orthogonal transforms Bound of the integration error: err QMC f d U α,γ e d,n In general, f d α,γ f d U α,γ Goal: Find U such that f d U α,γ grows slower in d than f d α,γ or is even bounded. LT-method (Imai & Tan 2007) Regression algorithm (I. & Leobacher 2012) Christian Irrgeher (JKU Linz) 16

46 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using forward method Christian Irrgeher (JKU Linz) 17

47 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using forward method Discrete problem f d (x) = exp( 1 d dj=1 x j ) Christian Irrgeher (JKU Linz) 17

48 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using forward method Discrete problem f d (x) = exp( 1 d dj=1 x j ) Hermite coefficients of f d : f d (k) = e d k /2 k! Christian Irrgeher (JKU Linz) 17

49 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using forward method Discrete problem f d (x) = exp( 1 d dj=1 x j ) Hermite coefficients of f d : f d (k) = e d k /2 k! f d 2 α,γ = k N d 0 r(α, γ, k) 1 f d (k) 2 exp(1 + 1 d dj=1 γ 1 j ) Christian Irrgeher (JKU Linz) 17

50 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using forward method Discrete problem f d (x) = exp( 1 d dj=1 x j ) Hermite coefficients of f d : f d (k) = e d k /2 k! f d 2 α,γ = k N d 0 r(α, γ, k) 1 f d (k) 2 exp(1 + 1 d dj=1 γ 1 j ) If γ j = j 2, ed,n is bounded by cn 1/2 with constant c > 0, But fd α,γ = O(e d ) Christian Irrgeher (JKU Linz) 17

51 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d construction using Brownian bridge Christian Irrgeher (JKU Linz) 18

52 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d construction Discrete problem (f d U)(x) = exp(x 1 ) using Brownian bridge Christian Irrgeher (JKU Linz) 18

53 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using Brownian bridge construction Discrete problem (f d U)(x) = exp(x 1 ) Hermite coefficients of f d U: e 1 f d U(k) = k1 if k! 2 =... = k d = 0 0 else Christian Irrgeher (JKU Linz) 18

54 Example Compute E(exp(B 1 )) Simulation of B on time grid 1 d, 2 d,..., d d using Brownian bridge construction Discrete problem (f d U)(x) = exp(x 1 ) Hermite coefficients of f d U: e 1 f d U(k) = k1 if k! 2 =... = k d = 0 0 else Norm is independent of d f d U 2 α,γ = k N d 0 r(α, γ, k) f d U(k) 2 = e + = e + γ 1 1 e2 c(α) k=1 γ 1 1 kα 1 e k 1! Christian Irrgeher (JKU Linz) 18

55 Hermite coefficients and orthogonal transforms Define: AU : L 2 (R d, ϕ) L 2 (R d, ϕ) by A U f = f U Hm := span{h k (x) : k = m} AU,m := A Hm U Christian Irrgeher (JKU Linz) 19

56 Hermite coefficients and orthogonal transforms Define: AU : L 2 (R d, ϕ) L 2 (R d, ϕ) by A U f = f U Hm := span{h k (x) : k = m} AU,m := A Hm U Then: AU is a Hilbert space automorphism of L 2 (R d, ϕ) AU,m is a Hilbert space automorphism of H m L 2 (R d, ϕ) = m 0 H m AU = m 0 A U,m Christian Irrgeher (JKU Linz) 19

57 Hermite coefficients and orthogonal transforms in H m H k (x) = k G(x, t) t k with exponential generating function G t=0 Christian Irrgeher (JKU Linz) 20

58 Hermite coefficients and orthogonal transforms in H m H k (x) = k G(x, t) t k with exponential generating function G t=0 For any (β 1,..., β m ) {1,..., d} m : t βm = G(Ux, t) = t β1 t=0 d ξ 1,...,ξ m=1 ( ) U βmξ m U β1 ξ 1 G(x, t) t ξm t ξ1 t=0 Christian Irrgeher (JKU Linz) 20

59 Hermite coefficients and orthogonal transforms in H m H k (x) = k G(x, t) t k with exponential generating function G t=0 For any (β 1,..., β m ) {1,..., d} m : t βm = G(Ux, t) = t β1 t=0 d ξ 1,...,ξ m=1 ( ) U βmξ m U β1 ξ 1 G(x, t) t ξm t ξ1 t=0 Many (β 1,..., β m ) correspond to the same multi-index k Christian Irrgeher (JKU Linz) 20

60 Hermite coefficients and orthogonal transforms in H m Space K m := l 2 ({1,..., d} m ) takes account of the order of differentiation Christian Irrgeher (JKU Linz) 21

61 Hermite coefficients and orthogonal transforms in H m Space K m := l 2 ({1,..., d} m ) takes account of the order of differentiation Isometries: Jm : H m K m J m : K m H m Christian Irrgeher (JKU Linz) 21

62 Hermite coefficients and orthogonal transforms in H m Space K m := l 2 ({1,..., d} m ) takes account of the order of differentiation Isometries: Jm : H m K m J m : K m H m H m J m A U,m H m J m K m U m K m Christian Irrgeher (JKU Linz) 21

63 Hermite coefficients and orthogonal transforms in L 2 L 2 (R d, ϕ) J A U L 2 (R d, ϕ) J K K m 0 U m with K = K m, m 0 J = J m, J = Jm. m 0 m 0 Christian Irrgeher (JKU Linz) 22

64 Hermite coefficients and orthogonal transforms in L 2 L 2 (R d, ϕ) J A U L 2 (R d, ϕ) J K K m 0 U m with K = K m, m 0 J = J m, J = Jm. m 0 m 0 Thus A = J ( m 0 U m) J Christian Irrgeher (JKU Linz) 22

65 Remarks and open problems Is A U well defined on the Korobov space? Other weight structures (choice of weights) Sobolev space setting (RKHS, tractability) Concrete point sets (worst case error analysis, tractability) Christian Irrgeher (JKU Linz) 23

66 Thank you for your attention Christian Irrgeher (JKU Linz) 24