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.

Statement

Let (S,d) be some complete separable metric space, and let be a stochastic process. Suppose that for all times T > 0, there exist positive constants such that for all. Then there exists a modification \tilde{X} of X that is a continuous process, i.e. a process such that Furthermore, the paths of \tilde{X} are locally \gamma-Hölder-continuous for every.

Example

In the case of Brownian motion on, the choice of constants \alpha = 4, \beta = 1, will work in the Kolmogorov continuity theorem. Moreover, for any positive integer m, the constants \alpha = 2m, \beta = m-1 will work, for some positive value of K that depends on n and m.