Kolmogorov extension theorem

In mathematics, the Kolmogorov extension 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.

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 $F_{i}\subseteq \mathbb {R} ^{n}$,

$\nu _{t_{\pi (1)}\dots t_{\pi (k)}}\left(F_{1}\times \dots \times F_{k}\right)=\nu _{t_{1}\dots t_{k}}\left(F_{\pi ^{-1}(1)}\times \dots \times F_{\pi ^{-1}(k)}\right);$

1. for all measurable sets $F_{i}\subseteq \mathbb {R} ^{n}$,

$\nu _{t_{1}\dots t_{k}}\left(F_{1}\times \dots \times F_{k}\right)=\nu _{t_{1}\dots t_{k}t_{k+1}}\left(F_{1}\times \dots \times F_{k}\times \mathbb {R} ^{n}\right).$

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 the $\nu _{t_{1}\dots t_{k}}$ as its finite-dimensional distributions.

Reference

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