Gaussian probability space
In probability theory particularly in the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables. Important examples include the classical or abstract Wiener space with some suitable collection of Gaussian random variables.[1][2]
Definition
A Gaussian probability space [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}}) }[/math] consists of
- a (complete) probability space [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math],
- a closed subspace [math]\displaystyle{ \mathcal{H}\subset L^2(\Omega,\mathcal{F},P) }[/math] called the Gaussian space such that all [math]\displaystyle{ X\in \mathcal{H} }[/math] are mean zero Gaussian variables. Their σ-algebra is denoted as [math]\displaystyle{ \mathcal{F}_{\mathcal{H}} }[/math].
- a σ-algebra [math]\displaystyle{ \mathcal{F}^{\perp}_{\mathcal{H}} }[/math] called the transverse σ-algebra which is defined through
- [math]\displaystyle{ \mathcal{F}=\mathcal{F}_{\mathcal{H}} \otimes \mathcal{F}^{\perp}_{\mathcal{H}}. }[/math][3]
Irreducibility
A Gaussian probability space is called irreducible if [math]\displaystyle{ \mathcal{F}=\mathcal{F}_{\mathcal{H}} }[/math]. Such spaces are denoted as [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H}) }[/math]. Non-irreducible spaces are used to work on subspaces or to extend a given probability space.[3] Irreducible Gaussian probability spaces are classified by the dimension of the Gaussian space [math]\displaystyle{ \mathcal{H} }[/math].[4]
Subspaces
A subspace [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H}_1,\mathcal{A}^{\perp}_{\mathcal{H}_1}) }[/math] of a Gaussian probability space [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}}) }[/math] consists of
- a closed subspace [math]\displaystyle{ \mathcal{H}_1\subset \mathcal{H} }[/math],
- a sub σ-algebra [math]\displaystyle{ \mathcal{A}^{\perp}_{\mathcal{H}_1}\subset \mathcal{F} }[/math] of transverse random variables such that [math]\displaystyle{ \mathcal{A}^{\perp}_{\mathcal{H}_1} }[/math] and [math]\displaystyle{ \mathcal{A}_{\mathcal{H}_1} }[/math] are independent, [math]\displaystyle{ \mathcal{A}=\mathcal{A}_{\mathcal{H}_1}\otimes \mathcal{A}^{\perp}_{\mathcal{H}_1} }[/math] and [math]\displaystyle{ \mathcal{A}\cap\mathcal{F}^{\perp}_{\mathcal{H}}=\mathcal{A}^{\perp}_{\mathcal{H}_1} }[/math].[3]
Example:
Let [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}}) }[/math] be a Gaussian probability space with a closed subspace [math]\displaystyle{ \mathcal{H}_1\subset \mathcal{H} }[/math]. Let [math]\displaystyle{ V }[/math] be the orthogonal complement of [math]\displaystyle{ \mathcal{H}_1 }[/math] in [math]\displaystyle{ \mathcal{H} }[/math]. Since orthogonality implies independence between [math]\displaystyle{ V }[/math] and [math]\displaystyle{ \mathcal{H}_1 }[/math], we have that [math]\displaystyle{ \mathcal{A}_V }[/math] is independent of [math]\displaystyle{ \mathcal{A}_{\mathcal{H}_1} }[/math]. Define [math]\displaystyle{ \mathcal{A}^{\perp}_{\mathcal{H}_1} }[/math] via [math]\displaystyle{ \mathcal{A}^{\perp}_{\mathcal{H}_1}:=\sigma(\mathcal{A}_V,\mathcal{F}^{\perp}_{\mathcal{H}})=\mathcal{A}_V \vee \mathcal{F}^{\perp}_{\mathcal{H}} }[/math].
Remark
For [math]\displaystyle{ G=L^2(\Omega,\mathcal{F}^{\perp}_{\mathcal{H}},P) }[/math] we have [math]\displaystyle{ L^2(\Omega,\mathcal{F},P)=L^2((\Omega,\mathcal{F}_{\mathcal{H}},P);G) }[/math].
Fundamental algebra
Given a Gaussian probability space [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}}) }[/math] one defines the algebra of cylindrical random variables
- [math]\displaystyle{ \mathbb{A}_{\mathcal{H}}=\{F=P(X_1,\dots,X_n):X_i\in \mathcal{H}\} }[/math]
where [math]\displaystyle{ P }[/math] is a polynomial in [math]\displaystyle{ \R[X_n,\dots,X_n] }[/math] and calls [math]\displaystyle{ \mathbb{A}_{\mathcal{H}} }[/math] the fundamental algebra. For any [math]\displaystyle{ p\lt \infty }[/math] it is true that [math]\displaystyle{ \mathbb{A}_{\mathcal{H}}\subset L^p(\Omega,\mathcal{F},P) }[/math].
For an irreducible Gaussian probability [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H}) }[/math] the fundamental algebra [math]\displaystyle{ \mathbb{A}_{\mathcal{H}} }[/math] is a dense set in [math]\displaystyle{ L^p(\Omega,\mathcal{F},P) }[/math] for all [math]\displaystyle{ p\in[1,\infty[ }[/math].[4]
Numerical and Segal model
An irreducible Gaussian probability [math]\displaystyle{ (\Omega,\mathcal{F},P,\mathcal{H}) }[/math] where a basis was chosen for [math]\displaystyle{ \mathcal{H} }[/math] is called a numerical model. Two numerical models are isomorphic if their Gaussian spaces have the same dimension.[4]
Given a separable Hilbert space [math]\displaystyle{ \mathcal{G} }[/math], there exists always a canoncial irreducible Gaussian probability space [math]\displaystyle{ \operatorname{Seg}(\mathcal{G}) }[/math] called the Segal model with [math]\displaystyle{ \mathcal{G} }[/math] as a Gaussian space.[5]
Literature
- Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
References
- ↑ Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
- ↑ Nualart, David (2013). The Malliavin calculus and related topics. New York: Springer. p. 3. doi:10.1007/978-1-4757-2437-0.
- ↑ 3.0 3.1 3.2 Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. pp. 4–5. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
- ↑ 4.0 4.1 4.2 Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. pp. 13–14. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
- ↑ Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. p. 16. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
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