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| In [[mathematics]], the '''integral representation theorem for classical Wiener space''' is a result in the fields of [[measure theory]] and [[stochastic processes|stochastic analysis]]. Essentially, it shows how to decompose a [[Function (mathematics)|function]] on [[classical Wiener space]] into the sum of its [[expected value]] and an [[Itō integral]].
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| ==Statement of the theorem==
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| 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 Itō 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
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| :<math>F(\sigma) = \int_{C_{0}} F(p) \, \mathrm{d} \gamma (p) + \int_{0}^{T} \alpha^{F} (\sigma)_{t} \, \mathrm{d} \sigma_{t}</math>
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| for <math>\gamma</math>-almost all <math>\sigma \in C_{0}</math>.
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| In the above,
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| * <math> \int_{C_{0}} F(p) \, \mathrm{d} \gamma (p) = \mathbb{E} [F]</math> is the expected value of <math>F</math>; and
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| * the integral <math>\int_{0}^{T} \cdots\, \mathrm{d} \sigma_{t}</math> is an Itō integral.
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| The proof of the integral representation theorem requires the [[Clark-Ocone theorem]] from the [[Malliavin calculus]].
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| ==Corollary: integral representation for an arbitrary probability space==
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| Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[probability space]]. Let <math>B : [0, T] \times \Omega \to \mathbb{R}</math> be a [[Brownian motion]] (i.e. a [[stochastic process]] whose law is [[Wiener measure]]). Let <math>\{ \mathcal{F}_{t} | 0 \leq t \leq T \}</math> be the natural [[Filtration (abstract algebra)|filtration]] of <math>\mathcal{F}</math> by the Brownian motion <math>B</math>:
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| ::<math>\mathcal{F}_{t} = \sigma \{ B_{s}^{-1} (A) | A \in \mathrm{Borel} (\mathbb{R}), 0 \leq s \leq t \}.</math>
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| Suppose that <math>f \in L^{2} (\Omega; \mathbb{R})</math> is <math>\mathcal{F}_{T}</math>-measurable. Then there is a unique Itō integrable process <math>a^{f} \in L^{2} (B)</math> such that
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| ::<math>f = \mathbb{E}[f] + \int_{0}^{T} a_{t}^{f} \, \mathrm{d} B_{t}</math> <math>\mathbb{P}</math>-almost surely.
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| ==References==
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| *Mao Xuerong. ''Stochastic differential equations and their applications.'' Chichester: Horwood. (1997)
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| [[Category:Measure theory]]
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| [[Category:Probability theorems]]
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| [[Category:Stochastic calculus]]
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| [[Category:Theorems in analysis]]
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