Introduction to dual quantization and first applications Gilles - PowerPoint PPT Presentation

Quantization and Cubature A Cubature formulae Quantization for Cubature Application as Cubature formula Assume that we have access to the Voronoi-Cell weights w i (Γ) := P ( X ∈ C i (Γ)) . ⇒ The computation of E F ( � X Γ ) for some Lipschitz continuous F : R d → R becomes straightforward: � n � � � E F ( � X Γ ) = E F x i 1 C i (Γ) ( X ) = w i (Γ) F ( x i ) . i =1 i =1 As a first error estimate, we clearly have | E F ( X ) − E F ( � X Γ ) | ≤ [ F ] Lip E � X − � X Γ � . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 9 / 49

Quantization and Cubature A Cubature formulae Quantization for Cubature PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 10 / 49

Quantization and Cubature A Cubature formulae Quantization for Cubature Second order rate If F ∈ C 1 Lip and the grid Γ is a stationary , i.e. X Γ = E ( X | � � X Γ ) , then a Taylor expansion yields | E F ( X ) − E F ( � X Γ ) | ≤ [ F ′ ] Lip · E � X − � X Γ � 2 . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 10 / 49

Quantization and Cubature A Cubature formulae Quantization for Cubature Second order rate If F ∈ C 1 Lip and the grid Γ is a stationary , i.e. X Γ = E ( X | � � X Γ ) , then a Taylor expansion yields | E F ( X ) − E F ( � X Γ ) | ≤ [ F ′ ] Lip · E � X − � X Γ � 2 . Furthermore, if F is convex, then Jensen’s inequality implies for stationary Γ E F ( � X Γ ) ≤ E F ( X ) . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 10 / 49

Quantization and Cubature Applications Further Applications Applications for optimal quantization grids PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 11 / 49

Quantization and Cubature Applications Further Applications Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 11 / 49

Quantization and Cubature Applications Further Applications Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle Obstacle Problems: Valuation of Bermudan and American options ([Bally/Pag` es ’03]) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 11 / 49

Quantization and Cubature Applications Further Applications Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle Obstacle Problems: Valuation of Bermudan and American options ([Bally/Pag` es ’03]) δ -Hedging for American options ([Bally/Pag` es/Printems ’05]) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 11 / 49

Quantization and Cubature Applications Further Applications Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle Obstacle Problems: Valuation of Bermudan and American options ([Bally/Pag` es ’03]) δ -Hedging for American options ([Bally/Pag` es/Printems ’05]) Optimal Stochastic Control problems, e.g. Pricing of Swing options ([Bronstein/Pag` es/W. ’09] and [Bardou/Bouthemy/Pag` es ’09]) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 11 / 49

Dual Quantization Idea Dual Quantization PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization Idea Do not map X ( ω ) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X ( ω ). PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization Idea Do not map X ( ω ) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X ( ω ). X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization Idea Do not map X ( ω ) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X ( ω ). X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization Idea Do not map X ( ω ) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X ( ω ). X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization Idea Do not map X ( ω ) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X ( ω ). X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization Idea Do not map X ( ω ) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X ( ω ). λ 1 − λ λ X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 12 / 49

Dual Quantization Idea Dual Quantization X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 13 / 49

Dual Quantization Idea Dual Quantization X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 13 / 49

Dual Quantization Idea Dual Quantization X ( ω ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 13 / 49

Dual Quantization Stationary Operators Ideas behind Dual Quantization Suppose that τ = { t 1 , . . . , t d +1 } ⊂ R d spans a d -simplex in R d , i.e. t 1 , . . . , t d +1 are affinely independent. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 14 / 49

Dual Quantization Stationary Operators Ideas behind Dual Quantization Suppose that τ = { t 1 , . . . , t d +1 } ⊂ R d spans a d -simplex in R d , i.e. t 1 , . . . , t d +1 are affinely independent. Moreover, let U ∼ U [0 , 1] be defined on some exogenous probability space (Ω 0 , S 0 , P 0 ). Denoting by λ ( ξ ) the barycentric coordinate of ξ ∈ conv { τ } , we define a dual quantization operator J U τ : conv { τ } → τ as d +1 � ξ �→ t i 1 � i − 1 � . i � � λ j ( ξ ) ≤ U < λ j ( ξ ) i =1 j =1 j =1 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 14 / 49

Dual Quantization Stationary Operators Ideas behind Dual Quantization Suppose that τ = { t 1 , . . . , t d +1 } ⊂ R d spans a d -simplex in R d , i.e. t 1 , . . . , t d +1 are affinely independent. Moreover, let U ∼ U [0 , 1] be defined on some exogenous probability space (Ω 0 , S 0 , P 0 ). Denoting by λ ( ξ ) the barycentric coordinate of ξ ∈ conv { τ } , we define a dual quantization operator J U τ : conv { τ } → τ as d +1 � ξ �→ t i 1 � i − 1 � . i � � λ j ( ξ ) ≤ U < λ j ( ξ ) i =1 j =1 j =1 This operator satisfies a mean preserving property: d +1 � � � J U λ i ( ξ ) · t i = ξ, ∀ ξ ∈ conv { τ } . E 0 τ ( ξ ) = (2) i =1 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 14 / 49

Dual Quantization Stationary Operators Ideas behind Dual Quantization Suppose that τ = { t 1 , . . . , t d +1 } ⊂ R d spans a d -simplex in R d , i.e. t 1 , . . . , t d +1 are affinely independent. Moreover, let U ∼ U [0 , 1] be defined on some exogenous probability space (Ω 0 , S 0 , P 0 ). Denoting by λ ( ξ ) the barycentric coordinate of ξ ∈ conv { τ } , we define a dual quantization operator J U τ : conv { τ } → τ as d +1 � ξ �→ t i 1 � i − 1 � . i � � λ j ( ξ ) ≤ U < λ j ( ξ ) i =1 j =1 j =1 This operator satisfies a mean preserving property: d +1 � � � J U λ i ( ξ ) · t i = ξ, ∀ ξ ∈ conv { τ } . E 0 τ ( ξ ) = (2) i =1 Similarly, we can construct such an operator for any triangulation on a grid Γ = { x 1 , . . . , x n } , so that (2) holds for any ξ ∈ conv { Γ } . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 14 / 49

Dual Quantization Stationary Operators Stationarity Motivated by this observation, we call a random splitting operator J Γ : Ω 0 × R d → Γ for a grid Γ ⊂ R d intrinsic stationary , if � � J Γ ( ξ ) ∀ ξ ∈ conv { Γ } . E 0 = ξ, PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 15 / 49

Dual Quantization Stationary Operators Stationarity Motivated by this observation, we call a random splitting operator J Γ : Ω 0 × R d → Γ for a grid Γ ⊂ R d intrinsic stationary , if � � J Γ ( ξ ) ∀ ξ ∈ conv { Γ } . E 0 = ξ, The deeper meaning of this definition is revealed by the following Proposition. Proposition J Γ is intrinsic stationary, if and only if it satisfies the stationarity condition � � J Γ ( Y ) | Y = Y E P ⊗ P 0 for any r.v. Y : (Ω , S , P ) → ( R d , B d ) with supp( P Y ) ⊂ conv { Γ } . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 15 / 49

Dual Quantization Stationary Operators Stationarity Motivated by this observation, we call a random splitting operator J Γ : Ω 0 × R d → Γ for a grid Γ ⊂ R d intrinsic stationary , if � � J Γ ( ξ ) ∀ ξ ∈ conv { Γ } . E 0 = ξ, The deeper meaning of this definition is revealed by the following Proposition. Proposition J Γ is intrinsic stationary, if and only if it satisfies the stationarity condition � � J Γ ( Y ) | Y = Y E P ⊗ P 0 for any r.v. Y : (Ω , S , P ) → ( R d , B d ) with supp( P Y ) ⊂ conv { Γ } . Note that this kind of stationarity now is very robust, since it holds by construction for any r.v. Y with support in Γ. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 15 / 49

Dual Quantization Stationary Operators Stationarity II As in the case of regular quantization, this kind of stationarity also yields a second order bound. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 16 / 49

Dual Quantization Stationary Operators Stationarity II As in the case of regular quantization, this kind of stationarity also yields a second order bound. Proposition Lip , Γ ⊂ R d and J Γ be intrinsic stationary. Then it holds (a) Let F ∈ C 1 for any r.v. Y ∈ L 2 ( P ) with supp( P Y ) ⊂ conv { Γ } , | E F ( Y ) − E F ( J Γ ( Y ) | ≤ [ F ′ ] Lip · E � Y − J Γ ( Y ) � 2 . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 16 / 49

Dual Quantization Stationary Operators Stationarity II As in the case of regular quantization, this kind of stationarity also yields a second order bound. Proposition Lip , Γ ⊂ R d and J Γ be intrinsic stationary. Then it holds (a) Let F ∈ C 1 for any r.v. Y ∈ L 2 ( P ) with supp( P Y ) ⊂ conv { Γ } , | E F ( Y ) − E F ( J Γ ( Y ) | ≤ [ F ′ ] Lip · E � Y − J Γ ( Y ) � 2 . (b) If F is convex, then Jensen’s inequality implies E F ( J Γ ( X )) ≥ E F ( X ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 16 / 49

Dual Quantization Definition Dual Quantization Question What is the best approximation, which can be achieved by an intrinsic stationary operator J Γ for a given grid Γ of size n ∈ N ? PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 17 / 49

Dual Quantization Definition Dual Quantization Question What is the best approximation, which can be achieved by an intrinsic stationary operator J Γ for a given grid Γ of size n ∈ N ? Problem: The grid Γ gives raise to many possible triangulations. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 17 / 49

Dual Quantization Definition Dual Quantization Question What is the best approximation, which can be achieved by an intrinsic stationary operator J Γ for a given grid Γ of size n ∈ N ? Problem: The grid Γ gives raise to many possible triangulations. We aim at selecting the triangulation with the lowest p -inertia i.e. to solve n � F p ( ξ ; Γ) = min λ i � ξ − x i � p ∀ ξ ∈ conv(Γ) , λ ∈ R n i =1 � ξ s.t. [ x 1 ... x n � 1 ... 1 ] λ = , λ ≥ 0 1 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 17 / 49

Dual Quantization Definition Dual Quantization Question What is the best approximation, which can be achieved by an intrinsic stationary operator J Γ for a given grid Γ of size n ∈ N ? Problem: The grid Γ gives raise to many possible triangulations. We aim at selecting the triangulation with the lowest p -inertia i.e. to solve n � F p ( ξ ; Γ) = min λ i � ξ − x i � p ∀ ξ ∈ conv(Γ) , λ ∈ R n i =1 � ξ s.t. [ x 1 ... x n � 1 ... 1 ] λ = , λ ≥ 0 1 For every ξ ∈ conv(Γ) we choose the best “triangle” in Γ which contains ξ . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 17 / 49

Dual Quantization Definition Dual Quantization Question What is the best approximation, which can be achieved by an intrinsic stationary operator J Γ for a given grid Γ of size n ∈ N ? Problem: The grid Γ gives raise to many possible triangulations. We aim at selecting the triangulation with the lowest p -inertia i.e. to solve n � F p ( ξ ; Γ) = min λ i � ξ − x i � p ∀ ξ ∈ conv(Γ) , λ ∈ R n i =1 � ξ s.t. [ x 1 ... x n � 1 ... 1 ] λ = , λ ≥ 0 1 For every ξ ∈ conv(Γ) we choose the best “triangle” in Γ which contains ξ . ⊲ The optimal p -th dual quantization error is then defined as � � d p E F p ( X ; Γ) : Γ ⊂ R d , | Γ | ≤ n n ( X ) = inf . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 17 / 49

Dual Quantization Definition Dual Quantization Optimality regions for F p ( ξ ; Γ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 18 / 49

Dual Quantization Definition Dual Quantization Optimality regions for F p ( ξ ; Γ) ⊲ To design the optimal dual quantization operator matching F p ( ξ ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 18 / 49

Dual Quantization Definition Dual Quantization Optimality regions for F p ( ξ ; Γ) ⊲ To design the optimal dual quantization operator matching F p ( ξ ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. ⊲ ( λ i ) 1 ≤ i ≤ n �→ min λ ∈ R n � n i =1 λ i � ξ − x i � p atteins a minimum (at least) � ξ s.t. [ x 1 ... x n � 1 ... 1 ] λ = , λ ≥ 0 1 at an extremal n -tuple λ ∗ ( ξ ) of the convex constraint set. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 18 / 49

Dual Quantization Definition Dual Quantization Optimality regions for F p ( ξ ; Γ) ⊲ To design the optimal dual quantization operator matching F p ( ξ ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. ⊲ ( λ i ) 1 ≤ i ≤ n �→ min λ ∈ R n � n i =1 λ i � ξ − x i � p atteins a minimum (at least) � ξ s.t. [ x 1 ... x n � 1 ... 1 ] λ = , λ ≥ 0 1 at an extremal n -tuple λ ∗ ( ξ ) of the convex constraint set. Therefore, I ∗ ( ξ ) := { i : λ ∗ i ( ξ ) > 0 } defines an affinely independent family ( x i ) i ∈ I ∗ ( ξ ) which can be completed into a Γ-valued affine basis. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 18 / 49

Dual Quantization Definition Dual Quantization Optimality regions for F p ( ξ ; Γ) ⊲ To design the optimal dual quantization operator matching F p ( ξ ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. ⊲ ( λ i ) 1 ≤ i ≤ n �→ min λ ∈ R n � n i =1 λ i � ξ − x i � p atteins a minimum (at least) � ξ s.t. [ x 1 ... x n � 1 ... 1 ] λ = , λ ≥ 0 1 at an extremal n -tuple λ ∗ ( ξ ) of the convex constraint set. Therefore, I ∗ ( ξ ) := { i : λ ∗ i ( ξ ) > 0 } defines an affinely independent family ( x i ) i ∈ I ∗ ( ξ ) which can be completed into a Γ-valued affine basis. � ξ ∈ R d : ∃ I ∗ ( ξ ) ⊂ I } , D I (Γ) = PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 18 / 49

Dual Quantization Definition Dual Quantization PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 19 / 49

Dual Quantization Definition Dual Quantization or equivalently in term of linear programming � � � ξ � � ξ ∈ R d : λ I = A − 1 i � ξ − x i � p = F p ( ξ ; Γ) λ I D I (Γ) = ≥ 0 and , I 1 i ∈ I where � � I ∈ I (Γ) = J ⊂ { 1 , . . . , n } : | J | = d + 1 , rk( A J ) = d + 1 � x 1 ... x n � and A I denotes the submatrix of whose columns are given by 1 ... 1 I . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 19 / 49

Dual Quantization Properties of Dual Quantization Quadratic Euclidean case PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization Quadratic Euclidean case In the case �·� = |·| 2 and p = 2, optimality regions are to Delaunay “triangles” in Γ, i.e. the sphere spanned by such a d -simplex contains no further point in its interior. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization Quadratic Euclidean case In the case �·� = |·| 2 and p = 2, optimality regions are to Delaunay “triangles” in Γ, i.e. the sphere spanned by such a d -simplex contains no further point in its interior. The following theorem is an extention of an important theorem by Rajan ([Rajan ’91]). Theorem Let �·� = |·| 2 , p = 2 , and Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d . (a) If I ∈ I (Γ) defines a Delaunay triangle (or d -simplex), then � ξ � λ I = A − 1 I 1 provides a solution to F p ( ξ ; Γ) for every ξ ∈ conv { x j : j ∈ I } i.e. D I (Γ) = conv { x j : j ∈ I } . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization Quadratic Euclidean case In the case �·� = |·| 2 and p = 2, optimality regions are to Delaunay “triangles” in Γ, i.e. the sphere spanned by such a d -simplex contains no further point in its interior. The following theorem is an extention of an important theorem by Rajan ([Rajan ’91]). Theorem Let �·� = |·| 2 , p = 2 , and Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d . (a) If I ∈ I (Γ) defines a Delaunay triangle (or d -simplex), then � ξ � λ I = A − 1 I 1 provides a solution to F p ( ξ ; Γ) for every ξ ∈ conv { x j : j ∈ I } i.e. D I (Γ) = conv { x j : j ∈ I } . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49 ˚

Dual Quantization Properties of Dual Quantization Optimal dual quantization operator For a Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d , PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization Optimal dual quantization operator For a Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d , choose a Borel partition ( C I (Γ)) I ∈I (Γ) of conv { Γ } such that C I (Γ) ⊂ D I (Γ) , PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization Optimal dual quantization operator For a Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d , choose a Borel partition ( C I (Γ)) I ∈I (Γ) of conv { Γ } such that C I (Γ) ⊂ D I (Γ) , let U ∼ U [0 , 1] on (Ω 0 , S 0 , P 0 ). PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization Optimal dual quantization operator For a Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d , choose a Borel partition ( C I (Γ)) I ∈I (Γ) of conv { Γ } such that C I (Γ) ⊂ D I (Γ) , let U ∼ U [0 , 1] on (Ω 0 , S 0 , P 0 ). The optimal dual quantization operator J ∗ Γ is defined as � k � � � J ∗ Γ ( ξ ) = x i · 1 � i − 1 � 1 C I (Γ) ( ξ ) . i λ I λ I � j ( ξ ) ≤ U< � j ( ξ ) i =1 I ∈I (Γ) j =1 j =1 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization Optimal dual quantization operator For a Γ = { x 1 , . . . , x n } ⊂ R d with aff . dim { Γ } = d , choose a Borel partition ( C I (Γ)) I ∈I (Γ) of conv { Γ } such that C I (Γ) ⊂ D I (Γ) , let U ∼ U [0 , 1] on (Ω 0 , S 0 , P 0 ). The optimal dual quantization operator J ∗ Γ is defined as � k � � � J ∗ Γ ( ξ ) = x i · 1 � i − 1 � 1 C I (Γ) ( ξ ) . i λ I λ I � j ( ξ ) ≤ U< � j ( ξ ) i =1 I ∈I (Γ) j =1 j =1 One easily checks that this operator is intrinsic stationary. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization Equivalence of optimal dual quantization PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization Equivalence of optimal dual quantization The operator J ∗ Γ then leads to the following characterizations of the optimal dual quantization error: PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization Equivalence of optimal dual quantization The operator J ∗ Γ then leads to the following characterizations of the optimal dual quantization error: Theorem ([Pag` es/W. ’10a]) Let X ∈ L p ( P ) and n ∈ N . Then d p n ( X ) = PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization Equivalence of optimal dual quantization The operator J ∗ Γ then leads to the following characterizations of the optimal dual quantization error: Theorem ([Pag` es/W. ’10a]) Let X ∈ L p ( P ) and n ∈ N . Then � E � X − J Γ ( X ) � p : J Γ : Ω 0 × R d → Γ is intrinsic stationary , d p n ( X ) = inf � supp( P X ) ⊂ conv { Γ } , | Γ | ≤ n PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization Equivalence of optimal dual quantization The operator J ∗ Γ then leads to the following characterizations of the optimal dual quantization error: Theorem ([Pag` es/W. ’10a]) Let X ∈ L p ( P ) and n ∈ N . Then � E � X − J Γ ( X ) � p : J Γ : Ω 0 × R d → Γ is intrinsic stationary , d p n ( X ) = inf � supp( P X ) ⊂ conv { Γ } , | Γ | ≤ n � Y � p : � E � X − � = inf Y is a r.v. on (Ω × Ω 0 , S ⊗ S 0 , P ⊗ P 0 ) , � | � Y (Ω × Ω 0 ) | ≤ n, E ( � Y | X ) = X . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Unbounded support Extension to unbounded support PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support Extension to unbounded support ∈ conv { Γ } , Since it is not possible to obtain intrinsic stationarity for ξ / we have to limit the claim for stationarity to a subset of supp( P X ) in order to extend the dual quantization problem to distributions with unbounded support. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support Extension to unbounded support ∈ conv { Γ } , Since it is not possible to obtain intrinsic stationarity for ξ / we have to limit the claim for stationarity to a subset of supp( P X ) in order to extend the dual quantization problem to distributions with unbounded support. We therefore drop the requirement supp( P X ) ⊂ conv { Γ } in above theorem PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support Extension to unbounded support ∈ conv { Γ } , Since it is not possible to obtain intrinsic stationarity for ξ / we have to limit the claim for stationarity to a subset of supp( P X ) in order to extend the dual quantization problem to distributions with unbounded support. We therefore drop the requirement supp( P X ) ⊂ conv { Γ } in above theorem and set � � ¯ E � X − J Γ ( X ) � p : J Γ is intrinsic stationary , | Γ | ≤ n d p n ( X ) = inf . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support Extension to unbounded support ∈ conv { Γ } , Since it is not possible to obtain intrinsic stationarity for ξ / we have to limit the claim for stationarity to a subset of supp( P X ) in order to extend the dual quantization problem to distributions with unbounded support. We therefore drop the requirement supp( P X ) ⊂ conv { Γ } in above theorem and set � � ¯ E � X − J Γ ( X ) � p : J Γ is intrinsic stationary , | Γ | ≤ n d p n ( X ) = inf . This means that we use a Nearest Neighbor projection beyond conv { Γ } while preserving stationarity in the interior of conv { Γ } . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Existence Existence of optimal dual quantizers PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 24 / 49

Dual Quantization Existence Existence of optimal dual quantizers Theorem ([Pag` es/W. ’10a]) (a) Let p > 1 and assume that P X has a compact support. Then for every n ≥ d + 1 optimal dual quantizers actually exist, i.e. the dual quantization problem d p n ( X ) attains its infimum. Moreover, d p n ( X ) is (strictly) decreasing to 0 as n → ∞ , if it does not vanish. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 24 / 49

Dual Quantization Existence Existence of optimal dual quantizers Theorem ([Pag` es/W. ’10a]) (a) Let p > 1 and assume that P X has a compact support. Then for every n ≥ d + 1 optimal dual quantizers actually exist, i.e. the dual quantization problem d p n ( X ) attains its infimum. Moreover, d p n ( X ) is (strictly) decreasing to 0 as n → ∞ , if it does not vanish. (b) Let p > 1 and assume that the distribution P X is strongly continuous. Then also optimal quantizers for ¯ d p n ( X ) exists and ¯ d p n ( X ) is (strictly) decreasing to 0 as n → ∞ , if it does not vanish. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 24 / 49

Dual Quantization Asymptotics Asymptotic behavior Theorem ([Pag` es/W. ’10b]) ( a ) Let X ∈ L p + ( R d ) and denote by ϕ the λ d -density of the absolutely continuous part of P X . PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 25 / 49

Dual Quantization Asymptotics Asymptotic behavior Theorem ([Pag` es/W. ’10b]) ( a ) Let X ∈ L p + ( R d ) and denote by ϕ the λ d -density of the absolutely continuous part of P X . Then �� � d + p d n →∞ n p/d · ¯ R d ϕ d/ ( d + p ) dλ d d p lim n ( X ) = Q d,p, �·� · � � n →∞ n p/d · d p U ([0 , 1] d ) where Q d,p, �·� = lim . n PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 25 / 49

Dual Quantization Asymptotics Asymptotic behavior Theorem ([Pag` es/W. ’10b]) ( a ) Let X ∈ L p + ( R d ) and denote by ϕ the λ d -density of the absolutely continuous part of P X . Then �� � d + p d n →∞ n p/d · ¯ R d ϕ d/ ( d + p ) dλ d d p lim n ( X ) = Q d,p, �·� · � � n →∞ n p/d · d p U ([0 , 1] d ) where Q d,p, �·� = lim . n � � ( b ) If d = 1 , Q d,p, �·� = 2 p +1 n →∞ n p/d · e p . If d ≥ 2 , ??? p +2 lim U ([0 , 1]) n PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 25 / 49

Dual Quantization Asymptotics Asymptotic behavior Sketch of the proof PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics Asymptotic behavior Sketch of the proof Prove existence of the limit for U ([0 , 1] d ) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics Asymptotic behavior Sketch of the proof Prove existence of the limit for U ([0 , 1] d ) Derive upper and lower bounds for piecewise constant densities (with compact support) on hypercubes PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics Asymptotic behavior Sketch of the proof Prove existence of the limit for U ([0 , 1] d ) Derive upper and lower bounds for piecewise constant densities (with compact support) on hypercubes Use Differentiation of measure to cover the general case (still compact support) PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics Asymptotic behavior Sketch of the proof Prove existence of the limit for U ([0 , 1] d ) Derive upper and lower bounds for piecewise constant densities (with compact support) on hypercubes Use Differentiation of measure to cover the general case (still compact support) Random dual quantization argument (so-called extended Pierce Lemma) to get the unbounded case. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Numerical computations 1,05 1,00 0,95 0,90 0,85 0,80 0,75 0,70 0,65 0,60 0,55 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 −0,05 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Figure: Dual Quantization for U ([0 , 1] 2 ) and n = 8 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 27 / 49

Dual Quantization Numerical computations 1,05 1,00 0,95 0,90 0,85 0,80 0,75 0,70 0,65 0,60 0,55 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 −0,05 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Figure: Dual Quantization for U ([0 , 1] 2 ) and n = 12 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 28 / 49

Dual Quantization Numerical computations 1,05 1,00 0,95 0,90 0,85 0,80 0,75 0,70 0,65 0,60 0,55 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 −0,05 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Figure: Dual Quantization for U ([0 , 1] 2 ) and n = 13 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 29 / 49

Dual Quantization Numerical computations 1,05 1,00 0,95 0,90 0,85 0,80 0,75 0,70 0,65 0,60 0,55 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 −0,05 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Figure: Dual Quantization for U ([0 , 1] 2 ) and n = 16 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 30 / 49

Dual Quantization Numerical computations 3,50 3,25 3,00 2,75 2,50 2,25 2,00 1,75 1,50 1,25 1,00 0,75 0,50 0,25 0,00 −0,25 −0,50 −0,75 −1,00 −1,25 −1,50 −1,75 −2,00 −2,25 −2,50 −2,75 −3,00 −3,25 −3,50 −3,5 −3,0 −2,5 −2,0 −1,5 −1,0 −0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 Figure: Dual Quantization for N (0 , I 2 ) and N = 250 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 31 / 49

Dual Quantization Numerical computations Figure: Joint Dual Quantization of the BM and its supremum, N = 250 PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 32 / 49