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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

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SLIDE 1

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener chaos approach for optimal prediction

Daniel Alpay1 Alon Kipnis 1

1Department of Mathematics

Ben-Gurion University of the Negev

May 2012

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SLIDE 2

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

slide-3
SLIDE 3

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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SLIDE 4

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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SLIDE 5

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Decomposition

Let H be a Gaussian Hilbert space, (each h ∈ H is a zero mean Gaussian random variable on the probability space (Ω, F, P) Denote by H⋄n the symmetric tensor power of H, then Γ(H)

∞

  • n=0

H⋄n = L2 (Ω, F(H), P) Γ(H) is the symmetric Fock space over H Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0

Xn(ω), Xn(ω) ∈ H⋄n

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SLIDE 6

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Decomposition

Let H be a Gaussian Hilbert space, (each h ∈ H is a zero mean Gaussian random variable on the probability space (Ω, F, P) Denote by H⋄n the symmetric tensor power of H, then Γ(H)

∞

  • n=0

H⋄n = L2 (Ω, F(H), P) Γ(H) is the symmetric Fock space over H Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0

Xn(ω), Xn(ω) ∈ H⋄n

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SLIDE 7

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Decomposition

Let H be a Gaussian Hilbert space, (each h ∈ H is a zero mean Gaussian random variable on the probability space (Ω, F, P) Denote by H⋄n the symmetric tensor power of H, then Γ(H)

∞

  • n=0

H⋄n = L2 (Ω, F(H), P) Γ(H) is the symmetric Fock space over H Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0

Xn(ω), Xn(ω) ∈ H⋄n

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SLIDE 8

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener-Hermite Chaos

For an orthogonal basis {ηn, n ∈ N} for H define Hα(ω)

∞

  • n=0

hαn (ηn) , α ∈ J where α is a multi index α = (α0, α1, ..., αr, 0, ...) and {hn(x), n ∈ N} are the Hermite polynomials {Hα, α ∈ J} is an orthonormal basis for L2 (Ω, F(H), P) The subset {Hα(ω), |α| = n} where |α| ∞

n=0 αn is an orthonormal basis for H⋄n

Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • |α|=n

fαHα(ω) =

  • α∈J

fαHα(ω)

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SLIDE 9

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener-Hermite Chaos

For an orthogonal basis {ηn, n ∈ N} for H define Hα(ω)

∞

  • n=0

hαn (ηn) , α ∈ J where α is a multi index α = (α0, α1, ..., αr, 0, ...) and {hn(x), n ∈ N} are the Hermite polynomials {Hα, α ∈ J} is an orthonormal basis for L2 (Ω, F(H), P) The subset {Hα(ω), |α| = n} where |α| ∞

n=0 αn is an orthonormal basis for H⋄n

Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • |α|=n

fαHα(ω) =

  • α∈J

fαHα(ω)

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SLIDE 10

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener-Hermite Chaos

For an orthogonal basis {ηn, n ∈ N} for H define Hα(ω)

∞

  • n=0

hαn (ηn) , α ∈ J where α is a multi index α = (α0, α1, ..., αr, 0, ...) and {hn(x), n ∈ N} are the Hermite polynomials {Hα, α ∈ J} is an orthonormal basis for L2 (Ω, F(H), P) The subset {Hα(ω), |α| = n} where |α| ∞

n=0 αn is an orthonormal basis for H⋄n

Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • |α|=n

fαHα(ω) =

  • α∈J

fαHα(ω)

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SLIDE 11

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener-Hermite Chaos

For an orthogonal basis {ηn, n ∈ N} for H define Hα(ω)

∞

  • n=0

hαn (ηn) , α ∈ J where α is a multi index α = (α0, α1, ..., αr, 0, ...) and {hn(x), n ∈ N} are the Hermite polynomials {Hα, α ∈ J} is an orthonormal basis for L2 (Ω, F(H), P) The subset {Hα(ω), |α| = n} where |α| ∞

n=0 αn is an orthonormal basis for H⋄n

Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • |α|=n

fαHα(ω) =

  • α∈J

fαHα(ω)

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SLIDE 12

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener chaos in terms of multiple Itˆ

  • integrals

Let H = sp

  • g(t)dB(t), g ∈ L2(R)
  • , where {B(t)} is a

Brownian motion If f(t1, ..., tn) ∈ L2(R) is symmetric then

  • Rn fndB⋄n n!

∞

−∞

tn

−∞

· · · t2

−∞

fn(...)dB(t) · · · dB(tn) is well defined and belongs to H⋄n Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • Rn fndB⋄n
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SLIDE 13

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener chaos in terms of multiple Itˆ

  • integrals

Let H = sp

  • g(t)dB(t), g ∈ L2(R)
  • , where {B(t)} is a

Brownian motion If f(t1, ..., tn) ∈ L2(R) is symmetric then

  • Rn fndB⋄n n!

∞

−∞

tn

−∞

· · · t2

−∞

fn(...)dB(t) · · · dB(tn) is well defined and belongs to H⋄n Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • Rn fndB⋄n
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SLIDE 14

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Wiener chaos in terms of multiple Itˆ

  • integrals

Let H = sp

  • g(t)dB(t), g ∈ L2(R)
  • , where {B(t)} is a

Brownian motion If f(t1, ..., tn) ∈ L2(R) is symmetric then

  • Rn fndB⋄n n!

∞

−∞

tn

−∞

· · · t2

−∞

fn(...)dB(t) · · · dB(tn) is well defined and belongs to H⋄n Each X ∈ L2 (Ω, F(H), P) has the decomposition X(ω) =

∞

  • n=0
  • Rn fndB⋄n
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SLIDE 15

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos expansion and measurability

Previous Results

Lemma [Holden, Øksendal, Ubøe and Zhang 1996 or Kuo1996 Lemma 13.11] Suppose Y(t) ∈ L2 (Ω, F(H), P) is a stochastic process with the chaos expansion Y(t) =

∞

  • n=0
  • Rn fndB⋄n =

=

∞

  • n=0

∞

−∞

tn

−∞

· · · t2

−∞

fn(..., tn, t)dB(t) · · · dB(tn) Then Y(t) is Ft-adapted if and only if supp fn(·, t) ⊂

  • x ∈ Rn

+ : xi ≤ t, i = 1, ..., n

  • for all n
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SLIDE 16

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Kolmogorov-Wiener Prediction Problem

Fix a stationary zero mean Gaussian process {x(t), t ∈ R} The classical prediction problem of Kolmogorov(1939,1941) and Wiener(1949): find ˆ x(T) E

  • x(T)|F−∞0

, E

  • (x(T) − ˆ

x(T))2 |F−∞0 (the distribution of x(T) at a future time T > 0 conditional on the “past” {x(t) : t ≤ 0} (F−∞0 is the sub-sigma-field generated by {x(t) : t ≤ 0})

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SLIDE 17

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Kolmogorov-Wiener Prediction Problem

Fix a stationary zero mean Gaussian process {x(t), t ∈ R} The classical prediction problem of Kolmogorov(1939,1941) and Wiener(1949): find ˆ x(T) E

  • x(T)|F−∞0

, E

  • (x(T) − ˆ

x(T))2 |F−∞0 (the distribution of x(T) at a future time T > 0 conditional on the “past” {x(t) : t ≤ 0} (F−∞0 is the sub-sigma-field generated by {x(t) : t ≤ 0})

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SLIDE 18

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Kolmogorov-Wiener prediction problem

Alternative formulation: find ˆ B∆(T) E

  • B∆(T)|F−∞0

where B∆(t) is the stationary increment process t

0 x(s)ds.

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SLIDE 19

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

The Goal of this Talk

Chaos Expansion Approach for Prediction Develop a chaos expansion such that each X ∈ L2 (Ω, F(H), P) would have the decomposition X =

  • α∈J−

fαHα +

  • α∈J+

fαHα, where Hα(ω) ∈

  • F−∞0,

α ∈ J−

  • F−∞0⊥

α ∈ J+

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SLIDE 20

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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SLIDE 21

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Trigonometric Isomorphism

Consider a stationary zero mean Gaussian process {x(t), t ∈ R} with spectral function ∆: E [x(t)x(s)] = ∞

−∞

ei(t−s)γd∆(γ) The above equation defines an isomorphism x(t) − → eiγt

  • f the Hilbert space generated by {x(t), t ∈ R} and

L2(d∆) The problem of projecting x(T), T > 0, onto sp {x(t), t ≤ 0} is translated into the projection of eiγT

  • nto sp
  • eiγt, t ≤ 0
  • )
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SLIDE 22

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Trigonometric Isomorphism

Consider a stationary zero mean Gaussian process {x(t), t ∈ R} with spectral function ∆: E [x(t)x(s)] = ∞

−∞

ei(t−s)γd∆(γ) The above equation defines an isomorphism x(t) − → eiγt

  • f the Hilbert space generated by {x(t), t ∈ R} and

L2(d∆) The problem of projecting x(T), T > 0, onto sp {x(t), t ≤ 0} is translated into the projection of eiγT

  • nto sp
  • eiγt, t ≤ 0
  • )
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SLIDE 23

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Trigonometric Isomorphism

Consider a stationary zero mean Gaussian process {x(t), t ∈ R} with spectral function ∆: E [x(t)x(s)] = ∞

−∞

ei(t−s)γd∆(γ) The above equation defines an isomorphism x(t) − → eiγt

  • f the Hilbert space generated by {x(t), t ∈ R} and

L2(d∆) The problem of projecting x(T), T > 0, onto sp {x(t), t ≤ 0} is translated into the projection of eiγT

  • nto sp
  • eiγt, t ≤ 0
  • )
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SLIDE 24

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Szeg¨

  • Criterion

Either ∞

−∞

log ∆′(γ) 1 + γ2 dγ > −∞ and E

  • x(T)|F−∞0

= x(T)

  • r else

∞

−∞

log ∆′(γ) 1 + γ2 dγ = −∞ and E

  • x(T)|F−∞0

= x(T) We assume ∞

−∞ log ∆′(γ) 1+γ2 dγ > −∞ and

∆(γ) = γ

−∞ ∆′(u)du

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SLIDE 25

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Szeg¨

  • Criterion

Either ∞

−∞

log ∆′(γ) 1 + γ2 dγ > −∞ and E

  • x(T)|F−∞0

= x(T)

  • r else

∞

−∞

log ∆′(γ) 1 + γ2 dγ = −∞ and E

  • x(T)|F−∞0

= x(T) We assume ∞

−∞ log ∆′(γ) 1+γ2 dγ > −∞ and

∆(γ) = γ

−∞ ∆′(u)du

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SLIDE 26

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Szeg¨

  • Criterion

Either ∞

−∞

log ∆′(γ) 1 + γ2 dγ > −∞ and E

  • x(T)|F−∞0

= x(T)

  • r else

∞

−∞

log ∆′(γ) 1 + γ2 dγ = −∞ and E

  • x(T)|F−∞0

= x(T) We assume ∞

−∞ log ∆′(γ) 1+γ2 dγ > −∞ and

∆(γ) = γ

−∞ ∆′(u)du

slide-27
SLIDE 27

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

definition and properties

f : C − → C analytic in the upper(lower) half plane belong to H2+ (H2−) if f2± sup

b≷0

  • |f(a + bi)|2da

1/2 < ∞ The map f − → limbց0 f(a + bi) identifies H2+ with

  • L2 [0, ∞), the set of functions g ∈ L2(R) whose inverse

fourier transform

∨

g has support in [0, ∞) H2− ∼ = L2 (−∞, 0]. L2(R) = H2+ ⊕ H2− (via the Plancherel identity)

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SLIDE 28

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

definition and properties

f : C − → C analytic in the upper(lower) half plane belong to H2+ (H2−) if f2± sup

b≷0

  • |f(a + bi)|2da

1/2 < ∞ The map f − → limbց0 f(a + bi) identifies H2+ with

  • L2 [0, ∞), the set of functions g ∈ L2(R) whose inverse

fourier transform

∨

g has support in [0, ∞) H2− ∼ = L2 (−∞, 0]. L2(R) = H2+ ⊕ H2− (via the Plancherel identity)

slide-29
SLIDE 29

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

definition and properties

f : C − → C analytic in the upper(lower) half plane belong to H2+ (H2−) if f2± sup

b≷0

  • |f(a + bi)|2da

1/2 < ∞ The map f − → limbց0 f(a + bi) identifies H2+ with

  • L2 [0, ∞), the set of functions g ∈ L2(R) whose inverse

fourier transform

∨

g has support in [0, ∞) H2− ∼ = L2 (−∞, 0]. L2(R) = H2+ ⊕ H2− (via the Plancherel identity)

slide-30
SLIDE 30

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

definition and properties

f : C − → C analytic in the upper(lower) half plane belong to H2+ (H2−) if f2± sup

b≷0

  • |f(a + bi)|2da

1/2 < ∞ The map f − → limbց0 f(a + bi) identifies H2+ with

  • L2 [0, ∞), the set of functions g ∈ L2(R) whose inverse

fourier transform

∨

g has support in [0, ∞) H2− ∼ = L2 (−∞, 0]. L2(R) = H2+ ⊕ H2− (via the Plancherel identity)

slide-31
SLIDE 31

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

an orthogonal basis

The functions en (γ) = 1 √π 1 1 − iγ 1 + iγ 1 − iγ n , n ∈ Z, form an orthonormal basis for L2(R) Moreover, sp {en, n ≥ 0} = H2+, sp {en, n < 0} = H2− The inverse fourier transform of the en, n ≥ 0, are the Laguerre functions:

∨

en = 1 √πn! dn dγn

  • e−iγx (i − γ)n

|γ=−i

slide-32
SLIDE 32

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

an orthogonal basis

The functions en (γ) = 1 √π 1 1 − iγ 1 + iγ 1 − iγ n , n ∈ Z, form an orthonormal basis for L2(R) Moreover, sp {en, n ≥ 0} = H2+, sp {en, n < 0} = H2− The inverse fourier transform of the en, n ≥ 0, are the Laguerre functions:

∨

en = 1 √πn! dn dγn

  • e−iγx (i − γ)n

|γ=−i

slide-33
SLIDE 33

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

an orthogonal basis

The functions en (γ) = 1 √π 1 1 − iγ 1 + iγ 1 − iγ n , n ∈ Z, form an orthonormal basis for L2(R) Moreover, sp {en, n ≥ 0} = H2+, sp {en, n < 0} = H2− The inverse fourier transform of the en, n ≥ 0, are the Laguerre functions:

∨

en = 1 √πn! dn dγn

  • e−iγx (i − γ)n

|γ=−i

slide-34
SLIDE 34

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

  • uter functions

h ∈ H2− is outer if and only if sp

  • eiγth∗(γ), t ≤ 0
  • = H2−

If log ∆′(γ)dγ 1 + γ2 > −∞ then ∆′ can be expressed as ∆′(γ) = |h(γ)|2 with h outer and h∗(γ) = h(−γ)

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SLIDE 35

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Hardy Spaces H2+/H2−

  • uter functions

h ∈ H2− is outer if and only if sp

  • eiγth∗(γ), t ≤ 0
  • = H2−

If log ∆′(γ)dγ 1 + γ2 > −∞ then ∆′ can be expressed as ∆′(γ) = |h(γ)|2 with h outer and h∗(γ) = h(−γ)

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Framework

Definition

Given a spectral function ∆, subject to d∆(γ)

1+γ2 < ∞, we

introduce the Hilbert space L∆ =

  • f ∈ L2(R) | f2

∆

∞

−∞

| f(γ)|2d∆(γ) < ∞

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Framework

Definition

Given a spectral function ∆, subject to d∆(γ)

1+γ2 < ∞, we

introduce the Hilbert space L∆ =

  • f ∈ L2(R) | f2

∆

∞

−∞

| f(γ)|2d∆(γ) < ∞

  • The isometry map f −

→ I(f) identifies L∆ with a Gaussian Hilbert space defined on (Ω, F, P) (F is taken to be minimal). So for f, g ∈ L∆, I(f) ∈ I(L∆) is a zero mean Gaussian random variable and E [I(f)I(g)] = (f, g)∆ ∞

−∞

ˆ f(γ)ˆ g∗(γ)d∆(γ)

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Framework

Cont.

For t ∈ R define B∆(t)

  • I(1[0,t]),

t ≥ 0 −I(1[t,0]), t < 0

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Framework

Cont.

For t ∈ R define B∆(t)

  • I(1[0,t]),

t ≥ 0 −I(1[t,0]), t < 0 B∆(t) is a stationary increments Gaussian process with E [B∆(t)B∆(s)] = ∞

−∞

1 − e−iγt γ 1 − eiγs γ d∆(γ)

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Framework

Cont.

For t ∈ R define B∆(t)

  • I(1[0,t]),

t ≥ 0 −I(1[t,0]), t < 0 B∆(t) is a stationary increments Gaussian process with E [B∆(t)B∆(s)] = ∞

−∞

1 − e−iγt γ 1 − eiγs γ d∆(γ) x(t) = ˙ B∆(t) is a stationary Gaussian process with spectral density ∆′. Namely, E [x(t)x(s)] = ∞

−∞

e−iγ(t−s)d∆(γ)

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Basis for I(L∆)

Definition

Let ∆′(γ) = |h(γ)|2 with h outer and h(−γ) = h∗(γ). Define ξn

∨

en h

  • (t)

n ∈ Z,

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Basis for I(L∆)

Definition

Let ∆′(γ) = |h(γ)|2 with h outer and h(−γ) = h∗(γ). Define ξn

∨

en h

  • (t)

n ∈ Z, Theorem The set {I(ξn), n ∈ Z} is an orthonormal basis for I(L∆) ⊂ L2 (Ω, F(I(L∆)), P). Moreover, E

  • I(ξn)|F−∞0

= I(ξn), n < 0 0, n ≥ 0 Thus the {I(ξn), n < 0} spans the past, and {I(ξn), n ≥ 0} spans its orthogonal complement

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

An Orthogonal Basis for I(L∆)

Proof of theorem

Proof. E [I(ξn)I(ξm)] = ∞

−∞

en h e∗

m

h∗ |h|2dγ = (en, em)L2(R) Denote by P the projection onto F−∞0. Let g denote the general sum c1eiγt1 + ... + cneiγtn with t1, ..., tn ≤ 0 E

  • (I(ξn) − P I(ξn))2

= inf

g en

h − g2

∆ =

inf

g

∞

−∞

|en h − g|2d∆(γ) = inf

g

∞

−∞

|en − h∗(γ)g(γ)|2dγ Since h is outer, the C.L.S. of

  • eiγth∗, t ≤ 0
  • is H2−
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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

An Orthogonal Basis for L2 (Ω, F, P)

Definition

Let J be the set of multi-indexes (..., α−1, α0, α1, ...) , αi ∈ N with at most finitely many non zero entries

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

An Orthogonal Basis for L2 (Ω, F, P)

Definition

Let J be the set of multi-indexes (..., α−1, α0, α1, ...) , αi ∈ N with at most finitely many non zero entries For α = (..., α−1, α0, α1, ...) ∈ J0 define Hα

∞

  • n=−∞

hαi (I(ξn)) where hk(x) is the kth Hermite polynomial

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

An Orthogonal Basis for L2 (Ω, F, P)

Definition

Let J be the set of multi-indexes (..., α−1, α0, α1, ...) , αi ∈ N with at most finitely many non zero entries For α = (..., α−1, α0, α1, ...) ∈ J0 define Hα

∞

  • n=−∞

hαi (I(ξn)) where hk(x) is the kth Hermite polynomial The set {Hα, α ∈ J} is an orthogonal basis for L2 (Ω, F, P) with E [HαHβ] = α! = αn!, α = β 0, α = β

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

An Orthogonal Basis for L2 (Ω, F, P)

main result

Theorem Hα is measurable with respect to the past F−∞0 if and only if αi = 0 for all i ≥ 0 (α ∈ J−)

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

An Orthogonal Basis for L2 (Ω, F, P)

main result

Theorem Hα is measurable with respect to the past F−∞0 if and only if αi = 0 for all i ≥ 0 (α ∈ J−) Proof. Consider L2 (Ω, F, P) as the symmetric Fock space of the Gaussian space I(L∆). Denote by ΓP the second quantization of the orthogonal projection P into the past F−∞0. Then E

  • Hα|F−∞0

=

∞

  • n=−∞

(ΓP)hαi (I(ξn)) =

∞

  • n=−∞

(P I(ξn))⋄αn =

−1

  • n=−∞

(I(ξn))⋄αn ·

∞

  • n=0

(0)⋄αn

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Corollary

For Y ∈ L2 (Ω, F, P) with Y =

α∈J fαHα the natural

decomposition Y = E

  • Y|F−∞0

+

  • Y − E
  • Y|F−∞0

is given by E

  • Y|F−∞0

=

  • α∈J−

fαHα, and Y − E

  • Y|F−∞0

=

  • α∈J\J−

fαHα. So that E

  • Y − E
  • Y|F−∞02

|F−∞0

  • =
  • α∈J\J−

f 2

αα!

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Kolmogorov-Wiener Prediction Problem

Example B∆(t) = I(1[0,t]) =

∞

  • n=−∞
  • ξn, 1[0,t]
  • ∆ I (ξn) =

∞

  • n=−∞
  • ξn, 1[0,t]
  • ∆ Hǫ

where ǫ(n) = (..., 0,

nth

  • 1 , 0, ...). It follows that

E

  • B∆(T)|F−∞0

=

−1

  • n=−∞
  • ξn, 1[0,T]
  • ∆ I (ξn)

The coprojection is ∞

n=0

  • ξn, 1[0,T]
  • ∆ I (ξn), so the

prediction error is

∞

  • n=0
  • ξn, 1[0,T]

2

∆

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Kolmogorov-Wiener Prediction Problem

Cont.

For f ∈ L∆, I(f) can be denoted by I(f) = ∞

−∞

f(t)dB∆(t).

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Kolmogorov-Wiener Prediction Problem

Cont.

For f ∈ L∆, I(f) can be denoted by I(f) = ∞

−∞

f(t)dB∆(t). Given the path {B∆(t), t ≤ 0}, I(ξn) for n < 0 can be computed by I(ξn) =

−∞

f(t)dB∆(t) I(f), f ∈ L∆ (interpreted as a Wick-Itˆ

  • -Hitsuida integral)
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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Expansion for Solutions of SPDE

Numerical solution of SPDE using Weiner Chaos Expansion is discussed for example in Luo(2006)

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Expansion for Solutions of SPDE

Numerical solution of SPDE using Weiner Chaos Expansion is discussed for example in Luo(2006) Main idea: derive an ordinary PDE for the deterministic chaos coefficients of the solution

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SLIDE 59

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Expansion for Solutions of SPDE

Numerical solution of SPDE using Weiner Chaos Expansion is discussed for example in Luo(2006) Main idea: derive an ordinary PDE for the deterministic chaos coefficients of the solution Our chaos expansion allows:

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SLIDE 60

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Expansion for Solutions of SPDE

Numerical solution of SPDE using Weiner Chaos Expansion is discussed for example in Luo(2006) Main idea: derive an ordinary PDE for the deterministic chaos coefficients of the solution Our chaos expansion allows:

1

Extension of this technique to systems disturbed by colored noises, where the stochastic integral is interpreted as a Wick-Itˆ

  • -Skorohod itntegral

∞

−∞

f(t)dB∆(t) I(f), f ∈ L∆

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Chaos Expansion for Solutions of SPDE

Numerical solution of SPDE using Weiner Chaos Expansion is discussed for example in Luo(2006) Main idea: derive an ordinary PDE for the deterministic chaos coefficients of the solution Our chaos expansion allows:

1

Extension of this technique to systems disturbed by colored noises, where the stochastic integral is interpreted as a Wick-Itˆ

  • -Skorohod itntegral

∞

−∞

f(t)dB∆(t) I(f), f ∈ L∆

2

Conditioning on the past (or future), to characterize system dynamics when past observations are available

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Stochastic PDE

Example

Example Consider the 1-D stochastic Burgers equation

  • du + 1

2 d dx (u2)dt = µ d2 dx2 u dt + σdB∆(t),

u(x, 0) = u0(x), u(0, t) = u(1, t), (t, x) ∈ (0, T] × [0, 1] (a unique solution u(t, x) with finite second moments exists if u0L2 < ∞). Write u(x, t) =

  • α∈J

fα(t, x)Hα

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Stochastic PDE

Example

Example

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Stochastic PDE

Example

Example The chaos coefficients satisfy the PDE system (Luo 2006) ∂ ∂t fα(x, t) + 1 2

  • γ∈J
  • 0≤β≤α

C(α, β, γ) ∂ ∂x (fα−β+γfβ+γ) (x, t) = µ ∂2 ∂x2 fα(x, t) + σ

∞

  • i=−∞

1αj=δi,j d dt (1t, ξn)∆

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Stochastic PDE

Example

Example The chaos coefficients satisfy the PDE system (Luo 2006) ∂ ∂t fα(x, t) + 1 2

  • γ∈J
  • 0≤β≤α

C(α, β, γ) ∂ ∂x (fα−β+γfβ+γ) (x, t) = µ ∂2 ∂x2 fα(x, t) + σ

∞

  • i=−∞

1αj=δi,j d dt (1t, ξn)∆ Assume that past noise realization is given. Conditioned solution is obtained by discarding multi-indexes α with non-zero positive indexes from the sums

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Modeling with the Wick Product

The Wick product in L2 (Ω, F, P) can be described by Hα ⋄ Hβ = Hα+β

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Modeling with the Wick Product

The Wick product in L2 (Ω, F, P) can be described by Hα ⋄ Hβ = Hα+β See [Holden, Øksendal, Ubøe and Zhang 1996] for stochastic PDE’s in which the Wick product replaces the ordinary product

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Modeling with the Wick Product

The Wick product in L2 (Ω, F, P) can be described by Hα ⋄ Hβ = Hα+β See [Holden, Øksendal, Ubøe and Zhang 1996] for stochastic PDE’s in which the Wick product replaces the ordinary product See [Alpay and Levanony 2007] and [Alpay, Levanony and Pinhas 2010] for linear systems theory with input-output relation defined through the Wick product

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Measurability Properties

α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi, . . .) Let X, Y ∈ L2 (Ω, F, P) If X, Y ∈ F−∞0 then X ⋄ Y ∈ F−∞0

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Measurability Properties

α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi, . . .) Let X, Y ∈ L2 (Ω, F, P) If X, Y ∈ F−∞0 then X ⋄ Y ∈ F−∞0 If X ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Measurability Properties

α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi, . . .) Let X, Y ∈ L2 (Ω, F, P) If X, Y ∈ F−∞0 then X ⋄ Y ∈ F−∞0 If X ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0 If X / ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Measurability Properties

α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi, . . .) Let X, Y ∈ L2 (Ω, F, P) If X, Y ∈ F−∞0 then X ⋄ Y ∈ F−∞0 If X ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0 If X / ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0 Note that it is possible that X · Y ∈ F−∞0 even is neither X, Y / ∈ F−∞0

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Measurability Properties

α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi, . . .) Let X, Y ∈ L2 (Ω, F, P) If X, Y ∈ F−∞0 then X ⋄ Y ∈ F−∞0 If X ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0 If X / ∈ F−∞0 and Y / ∈ F−∞0 then X ⋄ Y / ∈ F−∞0 Note that it is possible that X · Y ∈ F−∞0 even is neither X, Y / ∈ F−∞0 Amplification Chaos cannot be reversed in time

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Outline

1

Introduction Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

2

Main Results

3

Applications Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

4

Summary

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Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Summary

We started with a spectral function ∆(γ), subject to d∆(γ)

1+γ2 associated with a stationary Gaussian

stochastic process {x(t), t ∈ R}

slide-77
SLIDE 77

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Summary

We started with a spectral function ∆(γ), subject to d∆(γ)

1+γ2 associated with a stationary Gaussian

stochastic process {x(t), t ∈ R} We have constructed a basis for L2 (Ω, F, P) using the Hermite polynomials, the spectral decomposition d∆(γ) = |h(γ)|2dγ and the functions {en, n ∈ Z}

slide-78
SLIDE 78

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Summary

We started with a spectral function ∆(γ), subject to d∆(γ)

1+γ2 associated with a stationary Gaussian

stochastic process {x(t), t ∈ R} We have constructed a basis for L2 (Ω, F, P) using the Hermite polynomials, the spectral decomposition d∆(γ) = |h(γ)|2dγ and the functions {en, n ∈ Z} This basis admits a natural representation for the measurability of random variables in L2 (Ω, F, P) with respect to the past F−∞0 of {x(t), t ∈ R}, such that each chaos element is either measurable or independent

slide-79
SLIDE 79

Optimal Prediction D.Alpay and

  • A. Kipnis

Introduction

Wiener Chaos Trigonometric Isomorphism Approach to Kolmogorov-Wiener Prediction Problem

Main Results Applications

Prediction of Gaussian Processes Stochastic PDE A Note on the Wick Product

Summary

Summary

We started with a spectral function ∆(γ), subject to d∆(γ)

1+γ2 associated with a stationary Gaussian

stochastic process {x(t), t ∈ R} We have constructed a basis for L2 (Ω, F, P) using the Hermite polynomials, the spectral decomposition d∆(γ) = |h(γ)|2dγ and the functions {en, n ∈ Z} This basis admits a natural representation for the measurability of random variables in L2 (Ω, F, P) with respect to the past F−∞0 of {x(t), t ∈ R}, such that each chaos element is either measurable or independent It allows a solution for the Wiener-Kolmogorov prediction problem in terms of chaos expansion