# Kolmogorov continuity theorem

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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 of the theorem

Let be some 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 almost surely -Hölder continuous for every .

## Example

In the case of Brownian motion on , the choice of constants , , will work in the Kolmogorov continuity theorem.

## See Also

## References

- Øksendal, Bernt K. (2003).
*Stochastic Differential Equations: An Introduction with Applications*. Springer, Berlin. ISBN 3-540-04758-1. Theorem 2.2.3