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[edit]

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[edit]

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 .

See also[edit]

Kolmogorov extension theorem

References[edit]