- Kolmogorov continuity theorem
mathematics, the Kolmogorov continuity theorem is a theoremthat guarantees that a stochastic processthat satisfies certain constrains on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematicianAndrey Nikolaevich Kolmogorov.
tatement of the theorem
Let be a stochastic process, and suppose that for all times , there exist constants such that
for all . Then there exists a continuous version of , i.e. a process such that
* is sample continuous;
* for every time , .
In the case of
Brownian motionon , the choice of constants , , will work in the Kolmogorov continuity theorem.
* Theorem 2.2.3
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