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In [[Mathematics|mathematics]], the '''integral representation theorem for classical Wiener space''' is a result in the fields of measure theory and stochastic analysis. Essentially, it shows how to decompose a [[Function (mathematics)|function]] on [[Classical Wiener space|classical Wiener space]] into the sum of its [[Expected value|expected value]] and an Itô integral.
==Statement of the theorem==

Let <math>C_{0} ([0, T]; \mathbb{R})</math> (or simply <math>C_{0}</math> for short) be classical Wiener space with classical Wiener measure <math>\gamma</math>. If <math>F \in L^{2} (C_{0}; \mathbb{R})</math>, then there exists a unique Itô integrable process <math>\alpha^{F} : [0, T] \times C_{0} \to \mathbb{R}</math> (i.e. in <math>L^{2} (B)</math>, where <math>B</math> is canonical [[Brownian motion]]) such that

:<math>F(\sigma) = \int_{C_{0}} F(p) \, \mathrm{d} \gamma (p) + \int_{0}^{T} \alpha^{F} (\sigma)_{t} \, \mathrm{d} \sigma_{t}</math>

for <math>\gamma</math>-almost all <math>\sigma \in C_{0}</math>.

In the above,
* <math> \int_{C_{0}} F(p) \, \mathrm{d} \gamma (p) = \mathbb{E} [F]</math> is the expected value of <math>F</math>; and
* the integral <math>\int_{0}^{T} \cdots\, \mathrm{d} \sigma_{t}</math> is an Itô integral.

The proof of the integral representation theorem requires the Clark-Ocone theorem from the [[Malliavin calculus]].

==Corollary: integral representation for an arbitrary probability space==

Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[Probability space|probability space]]. Let <math>B : [0, T] \times \Omega \to \mathbb{R}</math> be a [[Brownian motion]] (i.e. a [[Stochastic process|stochastic process]] whose law is Wiener measure). Let <math>\{ \mathcal{F}_{t} | 0 \leq t \leq T \}</math> be the natural filtration of <math>\mathcal{F}</math> by the Brownian motion <math>B</math>:
::<math>\mathcal{F}_{t} = \sigma \{ B_{s}^{-1} (A) | A \in \mathrm{Borel} (\mathbb{R}), 0 \leq s \leq t \}.</math>
Suppose that <math>f \in L^{2} (\Omega; \mathbb{R})</math> is <math>\mathcal{F}_{T}</math>-measurable. Then there is a unique Itô integrable process <math>a^{f} \in L^{2} (B)</math> such that
::<math>f = \mathbb{E}[f] + \int_{0}^{T} a_{t}^{f} \, \mathrm{d} B_{t}</math> <math>\mathbb{P}</math>-almost surely.

==References==
*Mao Xuerong. ''Stochastic differential equations and their applications.'' Chichester: Horwood. (1997)

[[Category:Theorems in measure theory]]
[[Category:Probability theorems]]
[[Category:Stochastic calculus]]
[[Category:Theorems in analysis]]

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