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{{Short description|Mathematical theorem}}
In [[mathematics]], the '''Kolmogorov continuity theorem''' is a [[theorem]] that guarantees that a [[stochastic process]] that satisfies certain constraints on the [[moment (mathematics)|moments]] of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the [[Soviet Union|Soviet]] [[mathematician]] [[Andrey Kolmogorov|Andrey Nikolaevich Kolmogorov]].
==Statement==
Let <math>(S,d)</math> be some complete separable metric space, and let <math>X\colon [0, + \infty) \times \Omega \to S</math> be a stochastic process. Suppose that for all times <math>T > 0</math>, there exist positive constants <math>\alpha, \beta, K</math> such that
:<math>\mathbb{E} [d(X_t, X_s)^\alpha] \leq K | t - s |^{1 + \beta}</math>
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* {{cite book | author= Daniel W. Stroock, S. R. Srinivasa Varadhan | authorlink=Daniel W. Stroock, S. R. Srinivasa Varadhan | title=Multidimensional Diffusion Processes | publisher=Springer, Berlin | year=1997 | isbn=978-3-662-22201-0}} p. 51
[[Category:Theorems
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