Namespaces
Variants
Actions

Difference between revisions of "Wiener-Itô decomposition"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Automatically changed introduction)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 115 formulas, 111 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|part}}
 
''Itô–Wiener decomposition''
 
''Itô–Wiener decomposition''
  
 
An orthogonal decomposition of the [[Hilbert space|Hilbert space]] of square-integrable functions on a Gaussian space. It was first proved in 1938 by N. Wiener [[#References|[a6]]] in terms of homogeneous chaos (cf. also [[Wiener chaos decomposition|Wiener chaos decomposition]]). In 1951, K. Itô [[#References|[a1]]] defined multiple Wiener integrals to interpret homogeneous chaos and gave a different proof of the decomposition theorem.
 
An orthogonal decomposition of the [[Hilbert space|Hilbert space]] of square-integrable functions on a Gaussian space. It was first proved in 1938 by N. Wiener [[#References|[a6]]] in terms of homogeneous chaos (cf. also [[Wiener chaos decomposition|Wiener chaos decomposition]]). In 1951, K. Itô [[#References|[a1]]] defined multiple Wiener integrals to interpret homogeneous chaos and gave a different proof of the decomposition theorem.
  
Take an abstract Wiener space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300901.png" /> [[#References|[a3]]] (cf. also [[Wiener space, abstract|Wiener space, abstract]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300902.png" /> be the standard Gaussian measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300903.png" />. The abstract version of Wiener–Itô decomposition deals with a special orthogonal decomposition of the real Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300904.png" />.
+
Take an abstract Wiener space $( H , B )$ [[#References|[a3]]] (cf. also [[Wiener space, abstract|Wiener space, abstract]]). Let $\mu$ be the standard Gaussian measure on $B$. The abstract version of Wiener–Itô decomposition deals with a special orthogonal decomposition of the real Hilbert space $L ^ { 2 } ( \mu )$.
  
Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300905.png" /> defines a normal [[Random variable|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300906.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300907.png" /> with mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300908.png" /> and variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w1300909.png" /> [[#References|[a3]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009010.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009011.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009012.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009013.png" />-closure of the linear space spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009014.png" /> and random variables of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009018.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009019.png" /> is an increasing sequence of closed subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009020.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009021.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009023.png" /> be the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009025.png" />. The elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009026.png" /> are called homogeneous chaos of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009027.png" />. Obviously, the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009028.png" /> are orthogonal. Moreover, the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009029.png" /> is the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009031.png" />, namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009032.png" />.
+
Each $h \in H$ defines a normal [[Random variable|random variable]] $\tilde{h}$ on $B$ with mean $0$ and variance $| h | _ { H } ^ { 2 }$ [[#References|[a3]]]. Let $F _ { 0 } = \mathbf{R}$. For $n \geq 1$, let $F _ { n }$ be the $L ^ { 2 } ( \mu )$-closure of the linear space spanned by $1$ and random variables of the form $\tilde { h } _ { 1 } \ldots \tilde { h } _ { k }$ with $k \leq n$ and $h _ { j } \in H$ for $1 \leq j \leq k$. Then $\{ F _ { n } \} _ { n = 0 } ^ { \infty }$ is an increasing sequence of closed subspaces of $L ^ { 2 } ( \mu )$. Let $G _ { 0 } = \mathbf{R}$ and, for $n \geq 1$, let $G_n$ be the orthogonal complement of $F _ { n-1 } $ in $F _ { n }$. The elements in $G_n$ are called homogeneous chaos of degree $n$. Obviously, the spaces $G_n$ are orthogonal. Moreover, the Hilbert space $L ^ { 2 } ( \mu )$ is the direct sum of $G_n$ for $n \geq 0$, namely, $L ^ { 2 } ( \mu ) = \sum _ { n = 0 } ^ { \infty } G _ { n }$.
  
Fix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009033.png" />. To describe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009034.png" /> more precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009035.png" /> be the orthogonal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009036.png" /> onto the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009037.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009038.png" />, define
+
Fix $n \geq 1$. To describe $G_n$ more precisely, let $P_n$ be the orthogonal projection of $L ^ { 2 } ( \mu )$ onto the space $G_n$. For $h _ { 1 } \otimes \ldots \otimes h _ { n } \in H ^ { \otimes n }$, define
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009039.png" /></td> </tr></table>
+
\begin{equation*} \theta _ { n } ( h _ { 1 } \bigotimes \ldots \bigotimes h _ { n } ) = P _ { n } ( \tilde { h _ { 1 } } \ldots \tilde { h _ { n } } ). \end{equation*}
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009040.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009041.png" /> denotes the symmetric [[Tensor product|tensor product]]) and
+
Then $\theta _ { n } ( h _ { 1 } \otimes \ldots \otimes h _ { n } ) = \theta _ { n } ( h _ { 1 } \otimes^\wedge \ldots \otimes^\wedge \sim h _ { n } )$ (where $\otimes\hat{}$ denotes the symmetric [[Tensor product|tensor product]]) and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009042.png" /></td> </tr></table>
+
\begin{equation*} \left\| \theta _ { n } ( h _ { 1 } \bigotimes \ldots \bigotimes h _ { n } ) \right\| _ { L ^ { 2 } ( \mu ) } = \sqrt { n ! } \left| h _ { 1 } \widehat{\bigotimes} \ldots \widehat{\bigotimes} h _ { n } \right| _ { H ^{ \bigotimes  n }}. \end{equation*}
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009043.png" /> extends by continuity to a continuous [[Linear operator|linear operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009044.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009045.png" /> and is an [[Isometric mapping|isometric mapping]] (up to the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009046.png" />) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009047.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009048.png" />. Actually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009049.png" /> is surjective and so for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009050.png" />, there exists a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009053.png" />. Therefore, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009054.png" />, there exists a unique sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009055.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009056.png" /> such that
+
Thus, $\theta _ { n }$ extends by continuity to a continuous [[Linear operator|linear operator]] from $H ^{\otimes n}$ into $G_n$ and is an [[Isometric mapping|isometric mapping]] (up to the constant $\sqrt { n ! }$) from $H ^ { \widehat{\otimes} n }$ into $G_n$. Actually, $\theta _ { n }$ is surjective and so for any $\varphi \in G _ { n }$, there exists a unique $f \in H ^ { \hat{\otimes} n }$ such that $\theta _ { n } ( f ) = \varphi$ and $\| \varphi \| _ { L ^ { 2 } ( \mu ) } = \sqrt { n ! } | f | _ { H ^ { \otimes n } }$. Therefore, for any $\varphi \in L ^ { 2 } ( \mu )$, there exists a unique sequence $\{ f _ { n } \} _ { n = 0 } ^ { \infty }$ with $f _ { n } \in H ^ { \widehat{ \otimes } n }$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009057.png" /></td> </tr></table>
+
\begin{equation*} \varphi = \sum _ { n = 0 } ^ { \infty } \theta _ { n } ( f _ { n } ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009058.png" /></td> </tr></table>
+
\begin{equation*} \| \varphi \| _ { L ^ { 2 } ( \mu ) } ^ { 2 } = \sum _ { n = 0 } ^ { \infty } n ! | f _ { n } | ^ { 2 } _ { H ^ {\bigotimes n}}. \end{equation*}
  
 
This is the abstract version of the Wiener–Itô decomposition theorem [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]].
 
This is the abstract version of the Wiener–Itô decomposition theorem [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009059.png" />. Define a [[Norm|norm]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009060.png" /> by
+
Let $\Gamma ( H ) = \sum _ { n = 0 } ^ { \infty } H ^ { \widehat{\otimes} n }$. Define a [[Norm|norm]] on $\Gamma ( H )$ by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009061.png" /></td> </tr></table>
+
\begin{equation*} \| ( f _ { 0 } , f _ { 1 } , \ldots ) \| _ { \Gamma ( H ) } = \left( \sum _ { n = 0 } ^ { \infty } n ! |f _ { n } | _ { H^{\bigotimes n}  } ^ { 2 }  \right) ^ { 1 / 2 }. \end{equation*}
  
The Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009062.png" /> is called the Fock space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009063.png" /> (cf. also [[Fock space|Fock space]]). The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009065.png" /> are isomorphic under the [[Unitary operator|unitary operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009066.png" /> defined by
+
The Hilbert space $\Gamma ( H )$ is called the Fock space of $H$ (cf. also [[Fock space|Fock space]]). The spaces $\Gamma ( H )$ and $L ^ { 2 } ( \mu )$ are isomorphic under the [[Unitary operator|unitary operator]] $\Theta$ defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009067.png" /></td> </tr></table>
+
\begin{equation*} \Theta ( f _ { 0 } , f _ { 1 } , \ldots ) = \sum _ { n = 0 } ^ { \infty } \theta _ { n } ( f _ { n } ). \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009068.png" /> be an orthonormal basis (cf. also [[Orthogonal basis|Orthogonal basis]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009069.png" />. For any non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009070.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009071.png" />, define
+
Let $\{ e _ { k } : k \geq 1 \}$ be an orthonormal basis (cf. also [[Orthogonal basis|Orthogonal basis]]) for $H$. For any non-negative integers $n _ { 1 } , n _ { 2 } , \dots$ such that $n _ { 1 } + n _ { 2 } + \ldots = n$, define
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009072.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009072.png"/></td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009073.png" /> is the Hermite polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009074.png" /> (cf. also [[Hermite polynomials|Hermite polynomials]]). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009075.png" /> is an orthonormal basis for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009076.png" /> of homogeneous chaos of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009077.png" />. Hence the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009078.png" /> forms an orthonormal basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009079.png" />.
+
where $\mathcal{H} _ { k } ( x ) = ( - 1 ) ^ { n } e ^ { x ^ { 2 } / 2 } D _ { x } ^ { k } e ^ { - x ^ { 2 } / 2 }$ is the Hermite polynomial of degree $k$ (cf. also [[Hermite polynomials|Hermite polynomials]]). The set $\{ \varphi _ { n _ { 1 } , n _ { 2 } , \ldots } : n _ { j } \geq 0 , n _ { 1 } + n _ { 2 } + \ldots = n \}$ is an orthonormal basis for the space $G_n$ of homogeneous chaos of degree $n$. Hence the set $\{ \varphi _ { n _ { 1 } , n _ { 2 } , \ldots } : n _ { j } \geq 0 , n _ { 1 } + n _ { 2 } + \ldots = n , n \geq 0 \}$ forms an orthonormal basis for $L ^ { 2 } ( \mu )$.
  
Consider the classical Wiener space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009080.png" /> [[#References|[a3]]]. The Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009081.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009082.png" /> under the unitary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009084.png" />. The standard Gaussian measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009085.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009086.png" /> is the Wiener measure and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009088.png" />, is a [[Brownian motion|Brownian motion]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009089.png" />, the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009090.png" /> is exactly the [[Wiener integral|Wiener integral]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009091.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009093.png" />. The random variable
+
Consider the classical Wiener space $( C ^ { \prime } , C )$ [[#References|[a3]]]. The Hilbert space $C ^ { \prime }$ is isomorphic to $L ^ { 2 } ( [ 0,1 ] )$ under the unitary operator $\iota( g ) = g ^ { \prime }$, $g \in C ^ { \prime }$. The standard Gaussian measure $\mu$ on $C$ is the Wiener measure and $B ( t , \omega ) = \omega ( t )$, $\omega \in C$, is a [[Brownian motion|Brownian motion]]. For $g \in L ^ { 2 } ( [ 0,1 ] )$, the random variable $\widetilde{( \iota ^ { - 1 } g )}$ is exactly the [[Wiener integral|Wiener integral]] $I ( g ) = \int _ { 0 } ^ { 1 } g ( t ) d B ( t )$. Let $g _ { j } \in L ^ { 2 } ( [ 0,1 ] )$, $1 \leq j \leq n$. The random variable
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009094.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009094.png"/></td> </tr></table>
  
is a homogeneous chaos in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009095.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009096.png" /> extends by continuity to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009097.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009098.png" />,
+
is a homogeneous chaos in the space $G_n$. The mapping $ { I } _ { n }$ extends by continuity to the space $L ^ { 2 } ( [ 0,1 ] ^ { n } )$. For $g \in L ^ { 2 } ( [ 0,1 ] ^ { n } )$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w13009099.png" /></td> </tr></table>
+
\begin{equation*} I _ { n } ( g ) = \int _ { [ 0,1 ] ^ { n } } g ( t _ { 1 } , \ldots , t _ { n } ) d B ( t _ { 1 } ) \ldots d B ( t _ { n } ), \end{equation*}
  
where the right-hand side is a multiple Wiener integral of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090100.png" /> as defined by Itô in [[#References|[a1]]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090101.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090102.png" /> is the symmetrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090103.png" />.) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090104.png" /> there exists a unique sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090105.png" /> of symmetric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090106.png" /> such that
+
where the right-hand side is a multiple Wiener integral of order $n$ as defined by Itô in [[#References|[a1]]] and $\| I _ { n } ( g ) \| _ { L ^{ 2}  ( \mu ) } = \sqrt { n ! } | \hat{g} |  _ { L ^{ 2}  ( [ 0,1 ]  ^ { n } )}$ (where $\hat{g}$ is the symmetrization of $g$.) For any $\varphi \in L ^ { 2 } ( \mu )$ there exists a unique sequence $\{ g _ { n } \} _ { n = 0 } ^ { \infty }$ of symmetric functions $g _ { n } \in L ^ { 2 } ( [ 0,1 ] ^ { n } )$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090107.png" /></td> </tr></table>
+
\begin{equation*} \varphi = \sum _ { n = 0 } ^ { \infty } I _ { n } ( g _ { n } ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090108.png" /></td> </tr></table>
+
\begin{equation*} \| \varphi \| _ { L ^ { 2 } ( \mu ) } ^ { 2 } = \sum _ { n = 0 } ^ { \infty } n ! | g _ { n } | _ { L^2  ( [ 0,1 ] ^ { n } )  } ^ { 2 } . \end{equation*}
  
This is the Wiener–Itô decomposition theorem in terms of multiple Wiener integrals. An orthonormal basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090109.png" /> is given by the set
+
This is the Wiener–Itô decomposition theorem in terms of multiple Wiener integrals. An orthonormal basis for $L ^ { 2 } ( \mu )$ is given by the set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090110.png" /></td> </tr></table>
+
\begin{equation*} \frac { 1 } { \sqrt { n _ { 1 } ! n _ { 2 } ! \ldots } }. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090111.png" /></td> </tr></table>
+
\begin{equation*} .\mathcal{H} _ { n _ { 1 } } \left( \int _ { 0 } ^ { 1 } e _ { 1 } ( t ) d B ( t ) \right) \mathcal{H} _ { n _ { 2 } } \left( \int _ { 0 } ^ { 1 } e _ { 2 } ( t ) d B ( t ) \right) \ldots ,\; n _ { j } \geq 0 ,\; n _ { 1 } + n _ { 2 } + \ldots = n ,\; n \geq 0, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090112.png" /> is an orthonormal basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130090/w130090113.png" /> and the integrals are Wiener integrals.
+
where $\{ e _ { k } : k \geq 1 \}$ is an orthonormal basis for $L ^ { 2 } ( [ 0,1 ] )$ and the integrals are Wiener integrals.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Itô,  "Multiple Wiener integral"  ''J. Math. Soc. Japan'' , '''3'''  (1951)  pp. 157–169</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Kallianpur,  "Stochastic filtering theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.-H. Kuo,  "Gaussian measures in Banach spaces" , ''Lecture Notes in Mathematics'' , '''463''' , Springer  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.-H. Kuo,  "White noise distribution theory" , CRC  (1996)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Obata,  "White noise calculus and Fock space" , ''Lecture Notes in Mathematics'' , '''1577''' , Springer  (1994)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  N. Wiener,  "The homogeneous chaos"  ''Amer. J. Math.'' , '''60'''  (1938)  pp. 897–936</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  K. Itô,  "Multiple Wiener integral"  ''J. Math. Soc. Japan'' , '''3'''  (1951)  pp. 157–169</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G. Kallianpur,  "Stochastic filtering theory" , Springer  (1980)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H.-H. Kuo,  "Gaussian measures in Banach spaces" , ''Lecture Notes in Mathematics'' , '''463''' , Springer  (1975)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H.-H. Kuo,  "White noise distribution theory" , CRC  (1996)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  N. Obata,  "White noise calculus and Fock space" , ''Lecture Notes in Mathematics'' , '''1577''' , Springer  (1994)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  N. Wiener,  "The homogeneous chaos"  ''Amer. J. Math.'' , '''60'''  (1938)  pp. 897–936</td></tr></table>

Latest revision as of 17:43, 1 July 2020

Itô–Wiener decomposition

An orthogonal decomposition of the Hilbert space of square-integrable functions on a Gaussian space. It was first proved in 1938 by N. Wiener [a6] in terms of homogeneous chaos (cf. also Wiener chaos decomposition). In 1951, K. Itô [a1] defined multiple Wiener integrals to interpret homogeneous chaos and gave a different proof of the decomposition theorem.

Take an abstract Wiener space $( H , B )$ [a3] (cf. also Wiener space, abstract). Let $\mu$ be the standard Gaussian measure on $B$. The abstract version of Wiener–Itô decomposition deals with a special orthogonal decomposition of the real Hilbert space $L ^ { 2 } ( \mu )$.

Each $h \in H$ defines a normal random variable $\tilde{h}$ on $B$ with mean $0$ and variance $| h | _ { H } ^ { 2 }$ [a3]. Let $F _ { 0 } = \mathbf{R}$. For $n \geq 1$, let $F _ { n }$ be the $L ^ { 2 } ( \mu )$-closure of the linear space spanned by $1$ and random variables of the form $\tilde { h } _ { 1 } \ldots \tilde { h } _ { k }$ with $k \leq n$ and $h _ { j } \in H$ for $1 \leq j \leq k$. Then $\{ F _ { n } \} _ { n = 0 } ^ { \infty }$ is an increasing sequence of closed subspaces of $L ^ { 2 } ( \mu )$. Let $G _ { 0 } = \mathbf{R}$ and, for $n \geq 1$, let $G_n$ be the orthogonal complement of $F _ { n-1 } $ in $F _ { n }$. The elements in $G_n$ are called homogeneous chaos of degree $n$. Obviously, the spaces $G_n$ are orthogonal. Moreover, the Hilbert space $L ^ { 2 } ( \mu )$ is the direct sum of $G_n$ for $n \geq 0$, namely, $L ^ { 2 } ( \mu ) = \sum _ { n = 0 } ^ { \infty } G _ { n }$.

Fix $n \geq 1$. To describe $G_n$ more precisely, let $P_n$ be the orthogonal projection of $L ^ { 2 } ( \mu )$ onto the space $G_n$. For $h _ { 1 } \otimes \ldots \otimes h _ { n } \in H ^ { \otimes n }$, define

\begin{equation*} \theta _ { n } ( h _ { 1 } \bigotimes \ldots \bigotimes h _ { n } ) = P _ { n } ( \tilde { h _ { 1 } } \ldots \tilde { h _ { n } } ). \end{equation*}

Then $\theta _ { n } ( h _ { 1 } \otimes \ldots \otimes h _ { n } ) = \theta _ { n } ( h _ { 1 } \otimes^\wedge \ldots \otimes^\wedge \sim h _ { n } )$ (where $\otimes\hat{}$ denotes the symmetric tensor product) and

\begin{equation*} \left\| \theta _ { n } ( h _ { 1 } \bigotimes \ldots \bigotimes h _ { n } ) \right\| _ { L ^ { 2 } ( \mu ) } = \sqrt { n ! } \left| h _ { 1 } \widehat{\bigotimes} \ldots \widehat{\bigotimes} h _ { n } \right| _ { H ^{ \bigotimes n }}. \end{equation*}

Thus, $\theta _ { n }$ extends by continuity to a continuous linear operator from $H ^{\otimes n}$ into $G_n$ and is an isometric mapping (up to the constant $\sqrt { n ! }$) from $H ^ { \widehat{\otimes} n }$ into $G_n$. Actually, $\theta _ { n }$ is surjective and so for any $\varphi \in G _ { n }$, there exists a unique $f \in H ^ { \hat{\otimes} n }$ such that $\theta _ { n } ( f ) = \varphi$ and $\| \varphi \| _ { L ^ { 2 } ( \mu ) } = \sqrt { n ! } | f | _ { H ^ { \otimes n } }$. Therefore, for any $\varphi \in L ^ { 2 } ( \mu )$, there exists a unique sequence $\{ f _ { n } \} _ { n = 0 } ^ { \infty }$ with $f _ { n } \in H ^ { \widehat{ \otimes } n }$ such that

\begin{equation*} \varphi = \sum _ { n = 0 } ^ { \infty } \theta _ { n } ( f _ { n } ), \end{equation*}

\begin{equation*} \| \varphi \| _ { L ^ { 2 } ( \mu ) } ^ { 2 } = \sum _ { n = 0 } ^ { \infty } n ! | f _ { n } | ^ { 2 } _ { H ^ {\bigotimes n}}. \end{equation*}

This is the abstract version of the Wiener–Itô decomposition theorem [a2], [a4], [a5].

Let $\Gamma ( H ) = \sum _ { n = 0 } ^ { \infty } H ^ { \widehat{\otimes} n }$. Define a norm on $\Gamma ( H )$ by

\begin{equation*} \| ( f _ { 0 } , f _ { 1 } , \ldots ) \| _ { \Gamma ( H ) } = \left( \sum _ { n = 0 } ^ { \infty } n ! |f _ { n } | _ { H^{\bigotimes n} } ^ { 2 } \right) ^ { 1 / 2 }. \end{equation*}

The Hilbert space $\Gamma ( H )$ is called the Fock space of $H$ (cf. also Fock space). The spaces $\Gamma ( H )$ and $L ^ { 2 } ( \mu )$ are isomorphic under the unitary operator $\Theta$ defined by

\begin{equation*} \Theta ( f _ { 0 } , f _ { 1 } , \ldots ) = \sum _ { n = 0 } ^ { \infty } \theta _ { n } ( f _ { n } ). \end{equation*}

Let $\{ e _ { k } : k \geq 1 \}$ be an orthonormal basis (cf. also Orthogonal basis) for $H$. For any non-negative integers $n _ { 1 } , n _ { 2 } , \dots$ such that $n _ { 1 } + n _ { 2 } + \ldots = n$, define

where $\mathcal{H} _ { k } ( x ) = ( - 1 ) ^ { n } e ^ { x ^ { 2 } / 2 } D _ { x } ^ { k } e ^ { - x ^ { 2 } / 2 }$ is the Hermite polynomial of degree $k$ (cf. also Hermite polynomials). The set $\{ \varphi _ { n _ { 1 } , n _ { 2 } , \ldots } : n _ { j } \geq 0 , n _ { 1 } + n _ { 2 } + \ldots = n \}$ is an orthonormal basis for the space $G_n$ of homogeneous chaos of degree $n$. Hence the set $\{ \varphi _ { n _ { 1 } , n _ { 2 } , \ldots } : n _ { j } \geq 0 , n _ { 1 } + n _ { 2 } + \ldots = n , n \geq 0 \}$ forms an orthonormal basis for $L ^ { 2 } ( \mu )$.

Consider the classical Wiener space $( C ^ { \prime } , C )$ [a3]. The Hilbert space $C ^ { \prime }$ is isomorphic to $L ^ { 2 } ( [ 0,1 ] )$ under the unitary operator $\iota( g ) = g ^ { \prime }$, $g \in C ^ { \prime }$. The standard Gaussian measure $\mu$ on $C$ is the Wiener measure and $B ( t , \omega ) = \omega ( t )$, $\omega \in C$, is a Brownian motion. For $g \in L ^ { 2 } ( [ 0,1 ] )$, the random variable $\widetilde{( \iota ^ { - 1 } g )}$ is exactly the Wiener integral $I ( g ) = \int _ { 0 } ^ { 1 } g ( t ) d B ( t )$. Let $g _ { j } \in L ^ { 2 } ( [ 0,1 ] )$, $1 \leq j \leq n$. The random variable

is a homogeneous chaos in the space $G_n$. The mapping $ { I } _ { n }$ extends by continuity to the space $L ^ { 2 } ( [ 0,1 ] ^ { n } )$. For $g \in L ^ { 2 } ( [ 0,1 ] ^ { n } )$,

\begin{equation*} I _ { n } ( g ) = \int _ { [ 0,1 ] ^ { n } } g ( t _ { 1 } , \ldots , t _ { n } ) d B ( t _ { 1 } ) \ldots d B ( t _ { n } ), \end{equation*}

where the right-hand side is a multiple Wiener integral of order $n$ as defined by Itô in [a1] and $\| I _ { n } ( g ) \| _ { L ^{ 2} ( \mu ) } = \sqrt { n ! } | \hat{g} | _ { L ^{ 2} ( [ 0,1 ] ^ { n } )}$ (where $\hat{g}$ is the symmetrization of $g$.) For any $\varphi \in L ^ { 2 } ( \mu )$ there exists a unique sequence $\{ g _ { n } \} _ { n = 0 } ^ { \infty }$ of symmetric functions $g _ { n } \in L ^ { 2 } ( [ 0,1 ] ^ { n } )$ such that

\begin{equation*} \varphi = \sum _ { n = 0 } ^ { \infty } I _ { n } ( g _ { n } ), \end{equation*}

\begin{equation*} \| \varphi \| _ { L ^ { 2 } ( \mu ) } ^ { 2 } = \sum _ { n = 0 } ^ { \infty } n ! | g _ { n } | _ { L^2 ( [ 0,1 ] ^ { n } ) } ^ { 2 } . \end{equation*}

This is the Wiener–Itô decomposition theorem in terms of multiple Wiener integrals. An orthonormal basis for $L ^ { 2 } ( \mu )$ is given by the set

\begin{equation*} \frac { 1 } { \sqrt { n _ { 1 } ! n _ { 2 } ! \ldots } }. \end{equation*}

\begin{equation*} .\mathcal{H} _ { n _ { 1 } } \left( \int _ { 0 } ^ { 1 } e _ { 1 } ( t ) d B ( t ) \right) \mathcal{H} _ { n _ { 2 } } \left( \int _ { 0 } ^ { 1 } e _ { 2 } ( t ) d B ( t ) \right) \ldots ,\; n _ { j } \geq 0 ,\; n _ { 1 } + n _ { 2 } + \ldots = n ,\; n \geq 0, \end{equation*}

where $\{ e _ { k } : k \geq 1 \}$ is an orthonormal basis for $L ^ { 2 } ( [ 0,1 ] )$ and the integrals are Wiener integrals.

References

[a1] K. Itô, "Multiple Wiener integral" J. Math. Soc. Japan , 3 (1951) pp. 157–169
[a2] G. Kallianpur, "Stochastic filtering theory" , Springer (1980)
[a3] H.-H. Kuo, "Gaussian measures in Banach spaces" , Lecture Notes in Mathematics , 463 , Springer (1975)
[a4] H.-H. Kuo, "White noise distribution theory" , CRC (1996)
[a5] N. Obata, "White noise calculus and Fock space" , Lecture Notes in Mathematics , 1577 , Springer (1994)
[a6] N. Wiener, "The homogeneous chaos" Amer. J. Math. , 60 (1938) pp. 897–936
How to Cite This Entry:
Wiener-Itô decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener-It%C3%B4_decomposition&oldid=23147
This article was adapted from an original article by Hui-Hsiung Kuo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article