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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]]. | |||
==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 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 | |||
:<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 Itō 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]]. 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>: | |||
::<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 Itō 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:Measure theory]] | |||
[[Category:Probability theorems]] | |||
[[Category:Stochastic calculus]] | |||
[[Category:Theorems in analysis]] |
Revision as of 05:08, 21 May 2013
In 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 on classical Wiener space into the sum of its expected value and an Itō integral.
Statement of the theorem
Let (or simply for short) be classical Wiener space with classical Wiener measure . If , then there exists a unique Itō integrable process (i.e. in , where is canonical Brownian motion) such that
In the above,
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 be a probability space. Let be a Brownian motion (i.e. a stochastic process whose law is Wiener measure). Let be the natural filtration of by the Brownian motion :
Suppose that is -measurable. Then there is a unique Itō integrable process such that
References
- Mao Xuerong. Stochastic differential equations and their applications. Chichester: Horwood. (1997)