Kolmogorov extension theorem: Difference between revisions

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{{short description|Consistent set of finite-dimensional distributions will define a stochastic process}}

{{About|a theorem deals withon stochastic processes|a theorem deals withon extension of pre-measure|Hahn–Kolmogorov theorem}}

In [[mathematics]], the '''Kolmogorov extension theorem''' (also known as '''Kolmogorov existence theorem''', orthe '''Kolmogorov consistency theorem''' or the '''Daniell-Kolmogorov theorem''') is a [[theorem]] that guarantees that a suitably "consistent" collection of [[finite-dimensional distribution]]s will define a [[stochastic process]]. It is credited to the English mathematician [[Percy John Daniell]] and the [[Russia|Russian]] [[mathematician]] [[Andrey Kolmogorov|Andrey Nikolaevich Kolmogorov]].<ref>{{cite book | author=Øksendal, Bernt | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer |___location=Berlin | year=2003 |edition=Sixth | isbn=3-540-04758-1 |page=11 |url=https://books.google.com/books?id=VgQDWyihxKYC&pg=PA11 }}</ref>

==Statement of the theorem==

Let <math>T</math> denote some [[Interval (mathematics)|interval]] (thought of as "[[time]]"), and let <math>n \in \mathbb{N}</math>. For each <math>k \in \mathbb{N}</math> and finite [[sequence]] of distinct times <math>t_{1}, \dots, t_{k} \in T</math>, let <math>\nu_{t_{1} \dots t_{k}}</math> be a [[probability measure]] on <math>(\mathbb{R}^{n})^{k}.</math>. Suppose that these measures satisfy two consistency conditions:

1. for all [[permutation]]s <math>\pi</math> of <math>\{ 1, \dots, k \}</math> and measurable sets <math>F_{i} \subseteq \mathbb{R}^{n}</math>,

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2. for all measurable sets <math>F_{i} \subseteq \mathbb{R}^{n}</math>,<math>m \in \mathbb{N}</math>

:<math>\nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \nu_{t_{1} \dots t_{k}, t_{k + 1}, \dots , t_{k+m}} \left( F_{1} \times \dots \times F_{k} \times \underbrace{\mathbb{R}^{n} \times \dots \times \mathbb{R}^{n}}_{m} \right).</math>

Then there exists a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> and a stochastic process <math>X : T \times \Omega \to \mathbb{R}^{n}</math> such that

:<math>\nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \mathbb{P} \left( X_{t_{1}} \in F_{1}, \dots, X_{t_{k}} \in F_{k} \right)</math>

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In fact, it is always possible to take as the underlying probability space <math>\Omega = (\mathbb{R}^n)^T</math> and to take for <math>X</math> the canonical process <math>X\colon (t,Y) \mapsto Y_t</math>. Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure <math>\nu</math> on <math>(\mathbb{R}^n)^T</math> with marginals <math>\nu_{t_{1} \dots t_{k}}</math> for any finite collection of times <math>t_{1} \dots t_{k}</math>. Kolmogorov's extension theorem applies when <math>T</math> is uncountable, but the price to pay

for this level of generality is that the measure <math>\nu</math> is only defined on the product [[Σ-algebra#Product_σ-algebra|product σ-algebra]] of <math>(\mathbb{R}^n)^T</math>, which is not very rich.

==Explanation of the conditions==

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The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process <math>X</math>. Then the probability <math>\mathbb{P}(X_1 >0, X_2<0)</math> can be computed either as <math>\nu_{1,2}( \mathbb{R}_+ \times \mathbb{R}_-)</math> or as <math>\nu_{2,1}( \mathbb{R}_- \times \mathbb{R}_+)</math>. Hence, for the finite-dimensional distributions to be consistent, it must hold that

<math>\nu_{1,2}( \mathbb{R}_+ \times \mathbb{R}_-) = \nu_{2,1}( \mathbb{R}_- \times \mathbb{R}_+)</math>.

The first condition generalisesgeneralizes this obvious statement to hold for any number of time points <math>t_i</math>, and any control sets <math>F_i</math>.

Continuing the example, the second condition implies that <math>\mathbb{P}(X_1>0) = \mathbb{P}(X_1>0, X_2 \in \mathbb{R})</math>. Also this is a trivial condition that will be satisfied by any consistent family of finite-dimensional distributions.

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The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some <math>\mathbb{R}^{n}</math>-valued stochastic process, but the assumption that the state space be <math>\mathbb{R}^{n}</math> is unnecessary. In fact, any collection of measurable spaces together with a collection of [[inner regular measure]]s defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation. The formal statement of the general theorem is as follows.<ref>{{cite book |first=T. |last=Tao |authorlink=Terence Tao |title=An Introduction to Measure Theory |series=[[Graduate Studies in Mathematics]] |volume=126 |___location=Providence |publisher=American Mathematical Society |year=2011 |isbn=978-0-8218-6919-2 |page=195 |url=https://books.google.com/books?id=HoGDAwAAQBAJ&pg=PA195 }}</ref>

Let <math>T</math> be any set. Let <math> \{ (\Omega_t, \mathcal{F}_t) \}_{t \in T} </math> be some collection of measurable spaces, and for each <math> t \in T </math>, let <math> \tau_t</math> be a [[Hausdorff space|Hausdorff topology]] on <math> \Omega_t</math>. For each finite subset <math>J \subset T</math>, define

:<math>\Omega_J := \prod_{t\in J} \Omega_t</math>.

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* Aldrich, J. (2007) [http://www.emis.de/journals/JEHPS/Decembre2007/Aldrich.pdf "But you have to remember P.J.Daniell of Sheffield"] [http://www.emis.de/journals/JEHPS/indexang.html Electronic Journ@l for History of Probability and Statistics] December 2007.

[[Category:Theorems regardingabout stochastic processes]]