Wiener chaos approach for optimal Introduction Wiener Chaos - PowerPoint PPT Presentation

Optimal Prediction D.Alpay and A. Kipnis Wiener chaos approach for optimal Introduction Wiener Chaos prediction Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem Main Results Daniel Alpay 1 Alon Kipnis 1 Applications Prediction of Gaussian Processes Stochastic PDE 1 Department of Mathematics A Note on the Wick Product Ben-Gurion University of the Negev Summary May 2012

Outline Optimal Prediction Introduction 1 D.Alpay and A. Kipnis Wiener Chaos Introduction Trigonometric Isomorphism Approach to Wiener Chaos Kolmogorov-Wiener Prediction Problem Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem Main Results 2 Main Results Applications Applications 3 Prediction of Gaussian Processes Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product Stochastic PDE Summary A Note on the Wick Product Summary 4

Outline Optimal Prediction Introduction 1 D.Alpay and A. Kipnis Wiener Chaos Introduction Trigonometric Isomorphism Approach to Wiener Chaos Kolmogorov-Wiener Prediction Problem Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem Main Results 2 Main Results Applications Applications 3 Prediction of Gaussian Processes Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product Stochastic PDE Summary A Note on the Wick Product Summary 4

Outline Optimal Prediction Introduction 1 D.Alpay and A. Kipnis Wiener Chaos Introduction Trigonometric Isomorphism Approach to Wiener Chaos Kolmogorov-Wiener Prediction Problem Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem Main Results 2 Main Results Applications Applications 3 Prediction of Gaussian Processes Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product Stochastic PDE Summary A Note on the Wick Product Summary 4

Chaos Decomposition Optimal Prediction Let H be a Gaussian Hilbert space, (each h ∈ H is a D.Alpay and zero mean Gaussian random variable on the probability A. Kipnis space (Ω , F , P ) Introduction Denote by H ⋄ n the symmetric tensor power of H , then Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener ∞ � Prediction Problem H ⋄ n = L 2 (Ω , F ( H ) , P ) Γ( H ) � Main Results Applications n = 0 Prediction of Gaussian Processes Γ( H ) is the symmetric Fock space over H Stochastic PDE A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � X n ( ω ) ∈ H ⋄ n X ( ω ) = X n ( ω ) , n = 0

Chaos Decomposition Optimal Prediction Let H be a Gaussian Hilbert space, (each h ∈ H is a D.Alpay and zero mean Gaussian random variable on the probability A. Kipnis space (Ω , F , P ) Introduction Denote by H ⋄ n the symmetric tensor power of H , then Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener ∞ � Prediction Problem H ⋄ n = L 2 (Ω , F ( H ) , P ) Γ( H ) � Main Results Applications n = 0 Prediction of Gaussian Processes Γ( H ) is the symmetric Fock space over H Stochastic PDE A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � X n ( ω ) ∈ H ⋄ n X ( ω ) = X n ( ω ) , n = 0

Chaos Decomposition Optimal Prediction Let H be a Gaussian Hilbert space, (each h ∈ H is a D.Alpay and zero mean Gaussian random variable on the probability A. Kipnis space (Ω , F , P ) Introduction Denote by H ⋄ n the symmetric tensor power of H , then Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener ∞ � Prediction Problem H ⋄ n = L 2 (Ω , F ( H ) , P ) Γ( H ) � Main Results Applications n = 0 Prediction of Gaussian Processes Γ( H ) is the symmetric Fock space over H Stochastic PDE A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � X n ( ω ) ∈ H ⋄ n X ( ω ) = X n ( ω ) , n = 0

Wiener-Hermite Chaos Optimal Prediction For an orthogonal basis { η n , n ∈ N } for H define D.Alpay and ∞ � A. Kipnis H α ( ω ) � h α n ( η n ) , α ∈ J Introduction n = 0 Wiener Chaos Trigonometric where α is a multi index α = ( α 0 , α 1 , ..., α r , 0 , ... ) and Isomorphism Approach to { h n ( x ) , n ∈ N } are the Hermite polynomials Kolmogorov-Wiener Prediction Problem { H α , α ∈ J } is an orthonormal basis for L 2 (Ω , F ( H ) , P ) Main Results The subset Applications Prediction of { H α ( ω ) , | α | = n } Gaussian Processes where | α | � � ∞ Stochastic PDE n = 0 α n is an orthonormal basis for H ⋄ n A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � � � X ( ω ) = f α H α ( ω ) = f α H α ( ω ) n = 0 | α | = n α ∈ J

Wiener-Hermite Chaos Optimal Prediction For an orthogonal basis { η n , n ∈ N } for H define D.Alpay and ∞ � A. Kipnis H α ( ω ) � h α n ( η n ) , α ∈ J Introduction n = 0 Wiener Chaos Trigonometric where α is a multi index α = ( α 0 , α 1 , ..., α r , 0 , ... ) and Isomorphism Approach to { h n ( x ) , n ∈ N } are the Hermite polynomials Kolmogorov-Wiener Prediction Problem { H α , α ∈ J } is an orthonormal basis for L 2 (Ω , F ( H ) , P ) Main Results The subset Applications Prediction of { H α ( ω ) , | α | = n } Gaussian Processes where | α | � � ∞ Stochastic PDE n = 0 α n is an orthonormal basis for H ⋄ n A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � � � X ( ω ) = f α H α ( ω ) = f α H α ( ω ) n = 0 | α | = n α ∈ J

Wiener-Hermite Chaos Optimal Prediction For an orthogonal basis { η n , n ∈ N } for H define D.Alpay and ∞ � A. Kipnis H α ( ω ) � h α n ( η n ) , α ∈ J Introduction n = 0 Wiener Chaos Trigonometric where α is a multi index α = ( α 0 , α 1 , ..., α r , 0 , ... ) and Isomorphism Approach to { h n ( x ) , n ∈ N } are the Hermite polynomials Kolmogorov-Wiener Prediction Problem { H α , α ∈ J } is an orthonormal basis for L 2 (Ω , F ( H ) , P ) Main Results The subset Applications Prediction of { H α ( ω ) , | α | = n } Gaussian Processes where | α | � � ∞ Stochastic PDE n = 0 α n is an orthonormal basis for H ⋄ n A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � � � X ( ω ) = f α H α ( ω ) = f α H α ( ω ) n = 0 | α | = n α ∈ J

Wiener-Hermite Chaos Optimal Prediction For an orthogonal basis { η n , n ∈ N } for H define D.Alpay and ∞ � A. Kipnis H α ( ω ) � h α n ( η n ) , α ∈ J Introduction n = 0 Wiener Chaos Trigonometric where α is a multi index α = ( α 0 , α 1 , ..., α r , 0 , ... ) and Isomorphism Approach to { h n ( x ) , n ∈ N } are the Hermite polynomials Kolmogorov-Wiener Prediction Problem { H α , α ∈ J } is an orthonormal basis for L 2 (Ω , F ( H ) , P ) Main Results The subset Applications Prediction of { H α ( ω ) , | α | = n } Gaussian Processes where | α | � � ∞ Stochastic PDE n = 0 α n is an orthonormal basis for H ⋄ n A Note on the Wick Product Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition Summary ∞ � � � X ( ω ) = f α H α ( ω ) = f α H α ( ω ) n = 0 | α | = n α ∈ J

Wiener chaos in terms of multiple Itˆ o integrals Optimal Prediction �� � Let H = sp g ( t ) dB ( t ) , g ∈ L 2 ( R ) , where { B ( t ) } is a D.Alpay and A. Kipnis Brownian motion Introduction If f ( t 1 , ..., t n ) ∈ L 2 ( R ) is symmetric then Wiener Chaos Trigonometric � � ∞ � t n � t 2 Isomorphism Approach to R n f n dB ⋄ n � n ! Kolmogorov-Wiener · · · f n ( ... ) dB ( t ) · · · dB ( t n ) Prediction Problem −∞ −∞ −∞ Main Results Applications is well defined and belongs to H ⋄ n Prediction of Gaussian Processes Stochastic PDE Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition A Note on the Wick Product � Summary � ∞ R n f n dB ⋄ n X ( ω ) = n = 0

Wiener chaos in terms of multiple Itˆ o integrals Optimal Prediction �� � Let H = sp g ( t ) dB ( t ) , g ∈ L 2 ( R ) , where { B ( t ) } is a D.Alpay and A. Kipnis Brownian motion Introduction If f ( t 1 , ..., t n ) ∈ L 2 ( R ) is symmetric then Wiener Chaos Trigonometric � � ∞ � t n � t 2 Isomorphism Approach to R n f n dB ⋄ n � n ! Kolmogorov-Wiener · · · f n ( ... ) dB ( t ) · · · dB ( t n ) Prediction Problem −∞ −∞ −∞ Main Results Applications is well defined and belongs to H ⋄ n Prediction of Gaussian Processes Stochastic PDE Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition A Note on the Wick Product � Summary � ∞ R n f n dB ⋄ n X ( ω ) = n = 0

Wiener chaos in terms of multiple Itˆ o integrals Optimal Prediction �� � Let H = sp g ( t ) dB ( t ) , g ∈ L 2 ( R ) , where { B ( t ) } is a D.Alpay and A. Kipnis Brownian motion Introduction If f ( t 1 , ..., t n ) ∈ L 2 ( R ) is symmetric then Wiener Chaos Trigonometric � � ∞ � t n � t 2 Isomorphism Approach to R n f n dB ⋄ n � n ! Kolmogorov-Wiener · · · f n ( ... ) dB ( t ) · · · dB ( t n ) Prediction Problem −∞ −∞ −∞ Main Results Applications is well defined and belongs to H ⋄ n Prediction of Gaussian Processes Stochastic PDE Each X ∈ L 2 (Ω , F ( H ) , P ) has the decomposition A Note on the Wick Product � Summary � ∞ R n f n dB ⋄ n X ( ω ) = n = 0