Kolmogorov extension theorem

In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem or Kolmogorov consistency theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.[1]

Statement of the theorem

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

1. for all permutations \( \pi of \( \{ 1, \dots, k \} \) and measurable sets \( F_{i} \subseteq \mathbb{R}^{n} \) ,

\( \nu_{t_{\pi (1)} \dots t_{\pi (k)}} \left( F_{\pi (1)} \times \dots \times F_{ \pi(k)} \right) = \nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right); \)

2. for all measurable sets\( F_{i} \subseteq \mathbb{R}^{n},m \in \mathbb{N} \)

\( \nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \nu_{t_{1} \dots t_{k} t_{k + 1}, \dots , t_{k+m}} \left( F_{1} \times \dots \times F_{k} \times \mathbb{R}^{n} \times \dots \times \mathbb{R}^{n} \right). \)

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

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

for all \( t_{i} \in T, k \in \mathbb{N} and measurable sets \( F_{i} \subseteq \mathbb{R}^{n} \) , i.e.\( X has \nu_{t_{1} \dots t_{k}} \) as its finite-dimensional distributions relative to times \( t_{1} \dots t_{k} \) .

In fact, it is always possible to take as the underlying probability space\( \Omega = (\mathbb{R}^n)^T \) and to take for X the canonical process \( X\colon (t,Y) \mapsto Y_t \) . Therefore, an alternative way of stating Kolomogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure \( \nu \) on \( (\mathbb{R}^n)^T \) with marginals\( \nu_{t_{1} \dots t_{k}} \) for any finite collection of times \( t_{1} \dots t_{k} \) . The remarkable feature of Kolmogorov's extension theorem is that it does not require T to be countable, but the price to pay for this level of generality is that the measure \nu is only defined on the product σ-algebra of \( (\mathbb{R}^n)^T \) , which is not very rich.
Explanation of the conditions

The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process X. Then the probability \( \mathbb{P}(X_1 >0, X_2<0) \) can be computed either as \( \nu_{1,2}( \mathbb{R}_+ , \mathbb{R}_-) \) or as \( \nu_{2,1}( \mathbb{R}_- , \mathbb{R}_+) \) . Hence, for the finite-dimensional distributions to be consistent, it must hold that \( \nu_{1,2}( \mathbb{R}_+ , \mathbb{R}_-) = \nu_{2,1}( \mathbb{R}_- , \mathbb{R}_+ \) ). The first condition generalises this obvious statement to hold for any number of time points t\( _i \) , and any control sets \( F_i \) .

Continuing the example, the second condition implies that \( \mathbb{P}(X_1>0) = \mathbb{P}(X_1>0, X_2 \in \mathbb{R}) \) . Also this is a trivial statement that must be satisfied for any consistent family of finite-dimensional distributions.
Implications of the theorem

Since the two conditions are trivially satisfied for any stochastic process, the powerful statement of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.

The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions ("statistics") of the stochastic process. The theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose. In many situations, this means that one does not have to be explicit about what the probability space is. Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.

References

^ Øksendal, Bernt (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.

Mathematics Encyclopedia