Kolmogorov continuity theorem

In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version").

It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.

be some complete separable metric space, and let

be a stochastic process.

Suppose that for all times

, there exist positive constants

α , β ,

Then there exists a modification

β α

In the case of Brownian motion on

, the choice of constants

will work in the Kolmogorov continuity theorem.

Moreover, for any positive integer

will work, for some positive value of