Adapting quasi-Monte Carlo methods to simulation problems in - PowerPoint PPT Presentation

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

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

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

Numerical simulation – quasi-Monte Carlo (QMC) 1. Discretization ◮ E ( g ( B )) ≈ E ( g d ( B T d )) = E ( f d ( X 1 , . . . , X d )) =: I ( f d ) d , . . . , B d T ◮ ( X 1 , . . . , X d ) are independent N (0 , 1) Christian Irrgeher (JKU Linz) 3

Numerical simulation – quasi-Monte Carlo (QMC) 1. Discretization ◮ E ( g ( B )) ≈ E ( g d ( B T d )) = E ( f d ( X 1 , . . . , X d )) =: I ( f d ) d , . . . , B d T ◮ ( X 1 , . . . , X d ) are independent N (0 , 1) 2. QMC integration � N ◮ I ( f d ) ≈ 1 j =1 f d ( x j ) =: Q d,N ( f d ) N ◮ { x 1 , . . . , x N } ⊂ R d deterministic point set Christian Irrgeher (JKU Linz) 3

Numerical simulation – quasi-Monte Carlo (QMC) 1. Discretization ◮ E ( g ( B )) ≈ E ( g d ( B T d )) = E ( f d ( X 1 , . . . , X d )) =: I ( f d ) d , . . . , B d T ◮ ( X 1 , . . . , X d ) are independent N (0 , 1) 2. QMC integration � N ◮ I ( f d ) ≈ 1 j =1 f d ( x j ) =: Q d,N ( f d ) N ◮ { x 1 , . . . , x N } ⊂ R d deterministic point set ◮ Error of QMC algorithm Q d,N err := | E ( g ( B )) − Q d,N ( f d ) | Christian Irrgeher (JKU Linz) 3

Error estimate ◮ First estimate: � �� � � − Q d,N ( f d ) � � f d � f d � E ( g ( B )) − I � + � I � err ≤ discretization error integration error Christian Irrgeher (JKU Linz) 4

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

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

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

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

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

Gaussian measure and Hermite polynomials ⊤ x 2 π e − x 1 ◮ Density of the (standard) Gaussian measure ϕ ( x ) = √ 2 � ◮ L 2 ( R d , ϕ ) = { f : R d − R d f ( x ) 2 ϕ ( x ) d x < ∞} → R : f measurable , Christian Irrgeher (JKU Linz) 6

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

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

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

Hermite expansion ◮ Hermite expansion of f ∈ L 2 ( R d , ϕ ) � ˆ in L 2 f ( x ) = f ( k ) H k ( x ) k ∈ N d 0 � ◮ k -th Hermite coefficient ˆ f ( k ) = R d f ( x ) H k ( x ) ϕ ( x ) d x Christian Irrgeher (JKU Linz) 7

Hermite expansion ◮ Hermite expansion of f ∈ L 2 ( R d , ϕ ) � ˆ in L 2 f ( x ) = f ( k ) H k ( x ) k ∈ N d 0 � ◮ k -th Hermite coefficient ˆ f ( k ) = R d f ( x ) H k ( x ) ϕ ( x ) d x Theorem Let f ∈ L 2 ( R d , ϕ ) ∩ C ( R d ) and � 0 ˆ f ( k ) < ∞ . Then k ∈ N d � ˆ f ( x ) = f ( k ) H k ( x ) k ∈ N d 0 for all x ∈ R d . Christian Irrgeher (JKU Linz) 7

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

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 ◮ Introduce inner product: ∞ � r ( α, γ, k ) − 1 ˆ � f, g � α,γ := f ( k )ˆ g ( k ) k =0 Christian Irrgeher (JKU Linz) 8

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 ◮ Introduce inner product: ∞ � r ( α, γ, k ) − 1 ˆ � f, g � α,γ := f ( k )ˆ g ( k ) k =0 � ◮ Corresponding norm: � f � α,γ = � f, f � α,γ Christian Irrgeher (JKU Linz) 8

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 ◮ Introduce inner product: ∞ � r ( α, γ, k ) − 1 ˆ � f, g � α,γ := f ( k )ˆ g ( k ) k =0 � ◮ Corresponding norm: � f � α,γ = � f, f � α,γ ◮ Function space: H α,γ ( R , ϕ ) := { f ∈ L 2 ( R , ϕ ) ∩ C ( R ) : � f � α,γ < ∞} Christian Irrgeher (JKU Linz) 8

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

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 ◮ K α,γ ( · , y ) ∈ H α,γ ( R , ϕ ) ∀ y ∈ R ◮ � f, K α,γ ( · , y ) � α, γ = f ( y ) ∀ y ∈ R ∀ f ∈ H α,γ ( R , ϕ ) Christian Irrgeher (JKU Linz) 9

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 ◮ K α,γ ( · , y ) ∈ H α,γ ( R , ϕ ) ∀ y ∈ R ◮ � f, K α,γ ( · , y ) � α, γ = f ( y ) ∀ y ∈ R ∀ f ∈ H α,γ ( R , ϕ ) ◮ Series representation of the reproducing kernel ∞ � k − α H k ( x ) H k ( y ) K α,γ ( x, y ) = 1 + γ k =1 Christian Irrgeher (JKU Linz) 9

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

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

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

Korobov space of functions on R d ◮ For non-increasing weights γ = ( γ 1 , . . . , γ d ) H α, γ ( R d , ϕ ) := H α,γ 1 ( R , ϕ ) ⊗ . . . ⊗ H α,γ d ( R , ϕ ) ◮ Inner product: � f, g � α, γ = � 0 r ( α, γ , k ) − 1 ˆ f ( k )ˆ g ( k ) k ∈ N d with r ( α, γ , k ) = � d j =1 r ( α, γ j , k j ) Christian Irrgeher (JKU Linz) 11