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. 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 ) {displaystyle (S,d)} be some complete metric space, and let X : [ 0 , + ∞ ) × Ω → S {displaystyle X:[0,+infty ) imes Omega o S} be a stochastic process. Suppose that for all times T > 0 {displaystyle T>0} , there exist positive constants α , β , K {displaystyle alpha ,eta ,K} such that for all 0 ≤ s , t ≤ T {displaystyle 0leq s,tleq T} . Then there exists a modification X ~ {displaystyle { ilde {X}}} of X {displaystyle X} that is a continuous process, i.e. a process X ~ : [ 0 , + ∞ ) × Ω → S {displaystyle { ilde {X}}:[0,+infty ) imes Omega o S} such that Furthermore, the paths of X ~ {displaystyle { ilde {X}}} are locally γ {displaystyle gamma } -Hölder-continuous for every 0 < γ < β α {displaystyle 0<gamma <{ frac {eta }{alpha }}} . In the case of Brownian motion on R n {displaystyle mathbb {R} ^{n}} , the choice of constants α = 4 {displaystyle alpha =4} , β = 1 {displaystyle eta =1} , K = n ( n + 2 ) {displaystyle K=n(n+2)} will work in the Kolmogorov continuity theorem. Moreover for any positive integer m {displaystyle m} , the constants α = 2 m {displaystyle alpha =2m} , β = m − 1 {displaystyle eta =m-1} will work, for some positive value of K {displaystyle K} that depends on n {displaystyle n} and m {displaystyle m} . Kolmogorov extension theorem