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.
Let ( S , d ) be some metric space, and let X : [ 0 , + ∞ ) × Ω → S be a stochastic process. Suppose that for all times T > 0 , there exist positive constants α , β , K such that
E [ d ( X t , X s ) α ] ≤ K | t − s | 1 + β for all 0 ≤ s , t ≤ T . Then there exists a modification of X that is a continuous process, i.e. a process X ~ : [ 0 , + ∞ ) × Ω → S such that
X ~ is sample-continuous;for every time t ≥ 0 , P ( X t = X ~ t ) = 1. Furthermore, the paths of X ~ are almost surely locally γ -Hölder-continuous for every 0 < γ < β α .
In the case of Brownian motion on R n , the choice of constants α = 4 , β = 1 , K = n ( n + 2 ) will work in the Kolmogorov continuity theorem. Moreover for any positive integer m , the constants α = 2 m , β = m − 1 will work, for some positive value of K that depends on n and m .
