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.

Statement

Let $(S,d)$ be some complete separable metric space, and let $X\colon [0,+\infty )\times \Omega \to S$ be a stochastic process. Suppose that for all times $T>0$ , there exist positive constants $\alpha ,\beta ,K$ such that

\mathbb {E} [d(X_{t},X_{s})^{\alpha }]\leq K|t-s|^{1+\beta }

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

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

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

Example

In the case of Brownian motion on $\mathbb {R} ^{n}$ , the choice of constants $\alpha =4$ , $\beta =1$ , $K=n(n+2)$ will work in the Kolmogorov continuity theorem. Moreover, for any positive integer $m$ , the constants $\alpha =2m$ , $\beta =m-1$ will work, for some positive value of $K$ that depends on $n$ and $m$ .