Kolmogorov continuity theorem

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On stochastic processes

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.

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Statement

Let ( S , d ) {\displaystyle (S,d)} be some complete separable metric space, and let X : [ 0 , + ∞ ) × Ω → S {\displaystyle X\colon [0,+\infty )\times \Omega \to S} be a stochastic process. Suppose that for all times T > 0 {\displaystyle T>0} , there exist positive constants α , β , K {\displaystyle \alpha ,\beta ,K} such that

E [ d ( X t , X s ) α ] ≤ K | t − s | 1 + β {\displaystyle \mathbb {E} [d(X_{t},X_{s})^{\alpha }]\leq K|t-s|^{1+\beta }}

for all 0 ≤ s , t ≤ T {\displaystyle 0\leq s,t\leq T} . Then there exists a modification X ~ {\displaystyle {\tilde {X}}} of X {\displaystyle X} that is a continuous process, i.e. a process X ~ : [ 0 , + ∞ ) × Ω → S {\displaystyle {\tilde {X}}\colon [0,+\infty )\times \Omega \to S} such that

X ~ {\displaystyle {\tilde {X}}} is sample-continuous;
for every time t ≥ 0 {\displaystyle t\geq 0} , P ( X t = X ~ t ) = 1. {\displaystyle \mathbb {P} (X_{t}={\tilde {X}}_{t})=1.}

Furthermore, the paths of X ~ {\displaystyle {\tilde {X}}} are locally γ {\displaystyle \gamma } -Hölder-continuous for every 0 < γ < β α {\displaystyle 0<\gamma <{\tfrac {\beta }{\alpha }}} .

Example

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 \beta =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 \beta =m-1} will work, for some positive value of K {\displaystyle K} that depends on n {\displaystyle n} and m {\displaystyle m} .

Statement

Example

See also