Kolmogorov extension theorem
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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.
[edit] Statement of the theorem
Let T denote some interval (thought of as "time"), and let 
. For each 
and finite sequence of times 
, let 
be a probability measure on 
. Suppose that these measures satisfy two consistency conditions:
- for all permutations π of
and measurable sets 
,
- for all measurable sets ,
Then there exists a probability space 
and a stochastic process 
such that
for all 
, and measurable sets , i.e. X has the as its finite-dimensional distributions.
[edit] Reference
- Øksendal, Bernt (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.
