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