Kolmogorov extension theorem

Kolmogorov extension theorem

In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence 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. [cite book | author=Øksendal, Bernt | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer, Berlin | year=2003 | id=ISBN 3-540-04758-1]

tatement 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 u_{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},: u_{t_{pi (1)} dots t_{pi (k) left( F_{1} imes dots imes F_{k} ight) = u_{t_{1} dots t_{k left( F_{pi^{-1} (1)} imes dots imes F_{pi^{-1} (k)} ight);2. for all measurable sets F_{i} subseteq mathbb{R}^{n},m in mathbb{N}: u_{t_{1} dots t_{k left( F_{1} imes dots imes F_{k} ight) = u_{t_{1} dots t_{k} t_{k + 1}, dots , t_{k+m left( F_{1} imes dots imes F_{k} imes mathbb{R}^{n} imes dots imes mathbb{R}^{n} ight).Then there exists a probability space (Omega, mathcal{F}, mathbb{P}) and a stochastic process X : T imes Omega o mathbb{R}^{n} such that: u_{t_{1} dots t_{k left( F_{1} imes dots imes F_{k} ight) = mathbb{P} left( X_{t_{1 in F_{1}, dots, X_{t_{k in F_{k} ight)for all t_{i} in T, k in mathbb{N} and measurable sets F_{i} subseteq mathbb{R}^{n}, i.e. X has the u_{t_{1} dots t_{k as its finite-dimensional distributions.

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