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. It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.

Statement

Let be some complete separable metric space, and let be a stochastic process. Suppose that for all times, there exist positive constants such that
for all. Then there exists a modification of that is a continuous process, i.e. a process such that

is sample-continuous;
for every time,

Furthermore, the paths of are locally -Hölder-continuous for every.

Example

In the case of Brownian motion on, the choice of constants,, will work in the Kolmogorov continuity theorem. Moreover, for any positive integer, the constants, will work, for some positive value of that depends on and.