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.

References