wiener chaos approach for optimal predictionkipnisal/slides/chaos_approach...optimal prediction...

OptimalPrediction

D.Alpay andA. Kipnis

IntroductionWiener Chaos

TrigonometricIsomorphismApproach toKolmogorov-WienerPrediction Problem

Main Results

ApplicationsPrediction ofGaussian Processes

Stochastic PDE

A Note on the WickProduct

Summary

Wiener chaos approach for optimalprediction

Daniel Alpay1 Alon Kipnis 1

1Department of MathematicsBen-Gurion University of the Negev

May 2012

OptimalPrediction

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Outline

1 IntroductionWiener ChaosTrigonometric Isomorphism Approach toKolmogorov-Wiener Prediction Problem

2 Main Results

3 ApplicationsPrediction of Gaussian ProcessesStochastic PDEA Note on the Wick Product

4 Summary

OptimalPrediction

Main Results

Stochastic PDE

Summary

Outline

2 Main Results

4 Summary

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Stochastic PDE

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Outline

2 Main Results

4 Summary

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Chaos Decomposition

Let H be a Gaussian Hilbert space, (each h ∈ H is azero mean Gaussian random variable on the probabilityspace (Ω,F,P)

Denote by Hn the symmetric tensor power of H, then

Γ(H) ,∞⊕

Hn = L2 (Ω,F(H),P)

Γ(H) is the symmetric Fock space over HEach X ∈ L2 (Ω,F(H),P) has the decomposition

X (ω) =∞∑

Xn(ω), Xn(ω) ∈ Hn

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Chaos Decomposition

Γ(H) ,∞⊕

Hn = L2 (Ω,F(H),P)

X (ω) =∞∑

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Chaos Decomposition

Γ(H) ,∞⊕

Hn = L2 (Ω,F(H),P)

X (ω) =∞∑

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Wiener-Hermite Chaos

For an orthogonal basis ηn,n ∈ N for H define

Hα(ω) ,∞∏

hαn (ηn) , α ∈ J

where α is a multi index α = (α0, α1, ..., αr ,0, ...) andhn(x),n ∈ N are the Hermite polynomialsHα, α ∈ J is an orthonormal basis for L2 (Ω,F(H),P)

The subsetHα(ω), |α| = n

where |α| ,∑∞

n=0 αn is an orthonormal basis for Hn

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

X (ω) =∞∑

∑|α|=n

fαHα(ω) =∑α∈J

fαHα(ω)

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Hα(ω) ,∞∏

where |α| ,∑∞

X (ω) =∞∑

∑|α|=n

fαHα(ω)

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Hα(ω) ,∞∏

where |α| ,∑∞

X (ω) =∞∑

∑|α|=n

fαHα(ω)

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Hα(ω) ,∞∏

where |α| ,∑∞

X (ω) =∞∑

∑|α|=n

fαHα(ω)

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Wiener chaos in terms of multiple Ito integrals

Let H = sp∫

g(t)dB(t), g ∈ L2(R)

, where B(t) is aBrownian motionIf f (t1, ..., tn) ∈ L2(R) is symmetric then∫

RnfndBn , n!

∫ ∞−∞

∫ tn

−∞· · ·∫ t2

−∞fn(...)dB(t) · · · dB(tn)

is well defined and belongs to Hn

X (ω) =∞∑

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Let H = sp∫

RnfndBn , n!

∫ ∞−∞

∫ tn

−∞· · ·∫ t2

−∞fn(...)dB(t) · · · dB(tn)

X (ω) =∞∑

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Let H = sp∫

RnfndBn , n!

∫ ∞−∞

∫ tn

−∞· · ·∫ t2

−∞fn(...)dB(t) · · · dB(tn)

X (ω) =∞∑

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Chaos expansion and measurabilityPrevious Results

Lemma[Holden, Øksendal, Ubøe and Zhang 1996 or Kuo1996Lemma 13.11] Suppose Y (t) ∈ L2 (Ω,F(H),P) is astochastic process with the chaos expansion

Y (t) =∞∑

fndBn =

=∞∑

∫ ∞−∞

∫ 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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Kolmogorov-Wiener Prediction Problem

Fix a stationary zero mean Gaussian processx(t), t ∈ RThe classical prediction problem ofKolmogorov(1939,1941) and Wiener(1949): find

x(T ) , E[x(T )|F−∞0

[(x(T )− x(T ))2 |F−∞0

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

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Fix a stationary zero mean Gaussian processx(t), t ∈ RThe classical prediction problem ofKolmogorov(1939,1941) and Wiener(1949): find

x(T ) , E[x(T )|F−∞0

[(x(T )− x(T ))2 |F−∞0

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

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Kolmogorov-Wiener prediction problem

Alternative formulation: find

B∆(T ) , E[B∆(T )|F−∞0

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

∫ t0 x(s)ds.

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The Goal of this Talk

Chaos Expansion Approach for PredictionDevelop a chaos expansion such that eachX ∈ L2 (Ω,F(H),P) would have the decomposition

X =∑α∈J−

fαHα +∑α∈J+

fαHα,

Hα(ω) ∈

F−∞0, α ∈ J−(F−∞0)⊥ α ∈ J+

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Trigonometric Isomorphism

Consider a stationary zero mean Gaussian processx(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

of the Hilbert space generated by x(t), t ∈ R andL2(d∆)

The problem of projecting x(T ),T > 0, ontosp x(t), t ≤ 0 is translated into the projection of eiγT

onto sp

eiγt , t ≤ 0

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E [x(t)x(s)] =

∫ ∞−∞

ei(t−s)γd∆(γ)

x(t) −→ eiγt

onto sp

eiγt , t ≤ 0

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E [x(t)x(s)] =

∫ ∞−∞

ei(t−s)γd∆(γ)

x(t) −→ eiγt

onto sp

eiγt , t ≤ 0

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Szego Criterion

Either∫ ∞−∞

log ∆′(γ)

1 + γ2 dγ > −∞ and E[x(T )|F−∞0

]6= x(T )

or 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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Szego Criterion

Either∫ ∞−∞

log ∆′(γ)

1 + γ2 dγ > −∞ and E[x(T )|F−∞0

]6= x(T )

log ∆′(γ)

1 + γ2 dγ = −∞ and E[x(T )|F−∞0

]= x(T )

∆(γ) =∫ γ−∞∆′(u)du

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Szego Criterion

Either∫ ∞−∞

log ∆′(γ)

1 + γ2 dγ > −∞ and E[x(T )|F−∞0

]6= x(T )

log ∆′(γ)

1 + γ2 dγ = −∞ and E[x(T )|F−∞0

]= x(T )

∆(γ) =∫ γ−∞∆′(u)du

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Hardy Spaces H2+/H2−definition and properties

f : C −→ C analytic in the upper(lower) half planebelong to H2+ (H2−) if

‖f‖2± , supb≷0

(∫|f (a + bi)|2da

The map f −→ limb0 f (a + bi) identifies H2+ withL2 [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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‖f‖2± , supb≷0

(∫|f (a + bi)|2da

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‖f‖2± , supb≷0

(∫|f (a + bi)|2da

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‖f‖2± , supb≷0

(∫|f (a + bi)|2da

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Hardy Spaces H2+/H2−an orthogonal basis

The functions

en (γ) =1√π

11− iγ

(1 + iγ1− iγ

, 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 theLaguerre functions:

∨en =

1√πn!

(e−iγx (i − γ)n

)|γ=−i

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The functions

en (γ) =1√π

11− iγ

(1 + iγ1− iγ

, n ∈ Z,

Moreover,

sp en,n ≥ 0 = H2+, sp en,n < 0 = H2−

∨en =

1√πn!

)|γ=−i

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The functions

en (γ) =1√π

11− iγ

(1 + iγ1− iγ

, n ∈ Z,

Moreover,

sp en,n ≥ 0 = H2+, sp en,n < 0 = H2−

∨en =

1√πn!

)|γ=−i

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Hardy Spaces H2+/H2−outer functions

h ∈ H2− is outer if and only if

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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Hardy Spaces H2+/H2−outer functions

h ∈ H2− is outer if and only if

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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FrameworkDefinition

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

1+γ2 <∞, weintroduce the Hilbert space

L∆ =

f ∈ L2(R) | ‖f‖2∆ ,

∫ ∞−∞|f (γ)|2d∆(γ) <∞

The isometry map f −→ I(f ) identifies L∆ with aGaussian Hilbert space defined on (Ω,F,P) (F is takento be minimal). So for f ,g ∈ L∆, I(f ) ∈ I(L∆) is a zeromean Gaussian random variable and

E [I(f )I(g)] = (f ,g)∆ ,∫ ∞−∞

f (γ)g∗(γ)d∆(γ)

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FrameworkDefinition

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

1+γ2 <∞, weintroduce the Hilbert space

L∆ =

f ∈ L2(R) | ‖f‖2∆ ,

∫ ∞−∞|f (γ)|2d∆(γ) <∞

The isometry map f −→ I(f ) identifies L∆ with aGaussian Hilbert space defined on (Ω,F,P) (F is takento be minimal). So for f ,g ∈ L∆, I(f ) ∈ I(L∆) is a zeromean Gaussian random variable and

E [I(f )I(g)] = (f ,g)∆ ,∫ ∞−∞

f (γ)g∗(γ)d∆(γ)

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FrameworkCont.

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

1− eiγs

γd∆(γ)

x(t) = B∆(t) is a stationary Gaussian process withspectral density ∆′. Namely,

E [x(t)x(s)] =

∫ ∞−∞

e−iγ(t−s)d∆(γ)

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FrameworkCont.

For t ∈ R define

B∆(t) ,

I(1[0,t]), t ≥ 0−I(1[t ,0]), t < 0

E [B∆(t)B∆(s)] =

∫ ∞−∞

1− e−iγt

1− eiγs

γd∆(γ)

E [x(t)x(s)] =

∫ ∞−∞

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FrameworkCont.

For t ∈ R define

B∆(t) ,

I(1[0,t]), t ≥ 0−I(1[t ,0]), t < 0

E [B∆(t)B∆(s)] =

∫ ∞−∞

1− e−iγt

1− eiγs

γd∆(γ)

E [x(t)x(s)] =

∫ ∞−∞

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Basis for I(L∆)Definition

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

∨(en

)(t) n ∈ Z,

TheoremThe set I(ξn),n ∈ Z is an orthonormal basis forI(L∆) ⊂ L2 (Ω,F(I(L∆)),P). Moreover,

E[I(ξn)|F−∞0

I(ξn), n < 00, n ≥ 0

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

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Basis for I(L∆)Definition

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

∨(en

)(t) n ∈ Z,

TheoremThe set I(ξn),n ∈ Z is an orthonormal basis forI(L∆) ⊂ L2 (Ω,F(I(L∆)),P). Moreover,

E[I(ξn)|F−∞0

I(ξn), n < 00, n ≥ 0

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

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An Orthogonal Basis for I(L∆)Proof of theorem

Proof.

E [I(ξn)I(ξm)] =

∫ ∞−∞

he∗mh∗|h|2dγ = (en,em)L2(R)

Denote by P the projection onto F−∞0. Let g denote thegeneral sum c1eiγt1 + ...+ cneiγtn with t1, ..., tn ≤ 0

E[(I(ξn)− P I(ξn))2

]= inf

g‖en

h− g‖2∆ =

∫ ∞−∞|en

h− g|2d∆(γ) = inf

∫ ∞−∞|en − h∗(γ)g(γ)|2dγ

Since h is outer, the C.L.S. of

eiγth∗, t ≤ 0

is H2−

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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 6= β

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with at most finitely many non zero entriesFor α = (..., α−1, α0, α1, ...) ∈ J0 define

Hα ,∞∏

E [HαHβ] =α! =

∏αn!, α = β

0, α 6= β

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with at most finitely many non zero entriesFor α = (..., α−1, α0, α1, ...) ∈ J0 define

Hα ,∞∏

E [HαHβ] =α! =

∏αn!, α = β

0, α 6= β

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An Orthogonal Basis for L2 (Ω,F,P)main result

Theorem

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

Proof.Consider L2 (Ω,F,P) as the symmetric Fock space of theGaussian space I(L∆). Denote by ΓP the secondquantization of the orthogonal projection P into the pastF−∞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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An Orthogonal Basis for L2 (Ω,F,P)main result

Theorem

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

Proof.Consider L2 (Ω,F,P) as the symmetric Fock space of theGaussian space I(L∆). Denote by ΓP the secondquantization of the orthogonal projection P into the pastF−∞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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Corollary

For Y ∈ L2 (Ω,F,P) with Y =∑

α∈J fαHα the naturaldecomposition

Y = E[Y |F−∞0

Y − E[Y |F−∞0

])is given by

E[Y |F−∞0

]=∑α∈J−

fαHα,

andY − E

[Y |F−∞0

∑α∈J\J−

fαHα.

So that

Y − E[Y |F−∞0

])2|F−∞0

∑α∈J\J−

f 2αα!

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Example

B∆(t) = I(1[0,t]) =∞∑

n=−∞

(ξn,1[0,t]

I (ξn) =∞∑

n=−∞

(ξn,1[0,t]

Hε(n),

where ε(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 theprediction error is

∞∑n=0

(ξn,1[0,T ]

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Kolmogorov-Wiener Prediction ProblemCont.

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 becomputed by

I(ξn) =

−∞f (t)dB∆(t) , I(f ), f ∈ L∆

(interpreted as a Wick-Ito-Hitsuida integral)

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Kolmogorov-Wiener Prediction ProblemCont.

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 becomputed by

I(ξn) =

−∞f (t)dB∆(t) , I(f ), f ∈ L∆

(interpreted as a Wick-Ito-Hitsuida integral)

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Chaos Expansion for Solutions of SPDE

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

Main idea: derive an ordinary PDE for the deterministicchaos coefficients of the solutionOur chaos expansion allows:

1 Extension of this technique to systems disturbed bycolored noises, where the stochastic integral isinterpreted as a Wick-Ito-Skorohod itntegral∫ ∞

−∞f (t)dB∆(t) , I(f ), f ∈ L∆

2 Conditioning on the past (or future), to characterizesystem dynamics when past observations are available

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

Our chaos expansion allows:

−∞f (t)dB∆(t) , I(f ), f ∈ L∆

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Numerical solution of SPDE using Weiner ChaosExpansion is discussed for example in Luo(2006)Main idea: derive an ordinary PDE for the deterministicchaos coefficients of the solutionOur chaos expansion allows:

−∞f (t)dB∆(t) , I(f ), f ∈ L∆

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−∞f (t)dB∆(t) , I(f ), f ∈ L∆

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−∞f (t)dB∆(t) , I(f ), f ∈ L∆

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Stochastic PDEExample

ExampleConsider the 1-D stochastic Burgers equation

du + 12

ddx (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 existsif ‖u0‖L2 <∞). Write

u(x , t) =∑α∈J

fα(t , x)Hα

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Example

The chaos coefficients satisfy the PDE system (Luo2006)

∂tfα(x , t) +

∑γ∈J

∑0≤β≤α

C(α, β, γ)∂

∂x(fα−β+γ fβ+γ) (x , t)

= µ∂2

∂x2 fα(x , t) + σ∞∑

i=−∞1αj =δi,j

(1t , ξn)∆

Assume that past noise realization is given.Conditioned solution is obtained by discardingmulti-indexes α with non-zero positive indexes from thesums

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ExampleThe chaos coefficients satisfy the PDE system (Luo2006)

∂tfα(x , t) +

∑γ∈J

∑0≤β≤α

C(α, β, γ)∂

= µ∂2

∂x2 fα(x , t) + σ

∞∑i=−∞

1αj =δi,j

(1t , ξn)∆

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ExampleThe chaos coefficients satisfy the PDE system (Luo2006)

∂tfα(x , t) +

∑γ∈J

∑0≤β≤α

C(α, β, γ)∂

= µ∂2

∂x2 fα(x , t) + σ

∞∑i=−∞

1αj =δi,j

(1t , ξn)∆

OptimalPrediction

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Stochastic PDE

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2 Main Results

4 Summary

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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] forstochastic PDE’s in which the Wick product replacesthe ordinary productSee [Alpay and Levanony 2007] and [Alpay, Levanonyand Pinhas 2010] for linear systems theory withinput-output relation defined through the Wick product

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Hα Hβ = Hα+β

See [Holden, Øksendal, Ubøe and Zhang 1996] forstochastic PDE’s in which the Wick product replacesthe ordinary product

See [Alpay and Levanony 2007] and [Alpay, Levanonyand Pinhas 2010] for linear systems theory withinput-output relation defined through the Wick product

OptimalPrediction

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Stochastic PDE

Summary

Hα Hβ = Hα+β

See [Holden, Øksendal, Ubøe and Zhang 1996] forstochastic PDE’s in which the Wick product replacesthe ordinary productSee [Alpay and Levanony 2007] and [Alpay, Levanonyand Pinhas 2010] for linear systems theory withinput-output relation defined through the Wick product

OptimalPrediction

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Stochastic PDE

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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 isneither X ,Y /∈ F−∞0

AmplificationChaos cannot be reversed in time

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α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi , . . .)

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α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi , . . .)

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α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi , . . .)

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α + β = (. . . , α−1 + β−1, α0 + β0, α1 + β1, . . . , αi + βi , . . .)

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2 Main Results

4 Summary

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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 theHermite polynomials, the spectral decompositiond∆(γ) = |h(γ)|2dγ and the functions en,n ∈ ZThis basis admits a natural representation for themeasurability of random variables in L2 (Ω,F,P) withrespect to the past F−∞0 of x(t), t ∈ R, such thateach chaos element is either measurable orindependentIt allows a solution for the Wiener-Kolmogorovprediction problem in terms of chaos expansion

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stochastic process x(t), t ∈ RWe have constructed a basis for L2 (Ω,F,P) using theHermite polynomials, the spectral decompositiond∆(γ) = |h(γ)|2dγ and the functions en,n ∈ Z

This basis admits a natural representation for themeasurability of random variables in L2 (Ω,F,P) withrespect to the past F−∞0 of x(t), t ∈ R, such thateach chaos element is either measurable orindependentIt allows a solution for the Wiener-Kolmogorovprediction problem in terms of chaos expansion

OptimalPrediction

Main Results

Stochastic PDE

Summary

stochastic process x(t), t ∈ RWe have constructed a basis for L2 (Ω,F,P) using theHermite polynomials, the spectral decompositiond∆(γ) = |h(γ)|2dγ and the functions en,n ∈ ZThis basis admits a natural representation for themeasurability of random variables in L2 (Ω,F,P) withrespect to the past F−∞0 of x(t), t ∈ R, such thateach chaos element is either measurable orindependent

It allows a solution for the Wiener-Kolmogorovprediction problem in terms of chaos expansion

OptimalPrediction

Main Results

Stochastic PDE

Summary

stochastic process x(t), t ∈ RWe have constructed a basis for L2 (Ω,F,P) using theHermite polynomials, the spectral decompositiond∆(γ) = |h(γ)|2dγ and the functions en,n ∈ ZThis basis admits a natural representation for themeasurability of random variables in L2 (Ω,F,P) withrespect to the past F−∞0 of x(t), t ∈ R, such thateach chaos element is either measurable orindependentIt allows a solution for the Wiener-Kolmogorovprediction problem in terms of chaos expansion

wiener chaos approach for optimal predictionkipnisal/slides/chaos_approach...optimal prediction...

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