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 of the theorem

Let $(S,d)$ be some complete metric space such that d is ${\mathcal {B}}(S)\otimes {\mathcal {B}}(S)$ measurable, and let $X:[0,+\infty )\times \Omega \to S$ 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 ${\tilde {X}}:[0,+\infty )\times \Omega \to S$ 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 $m$.

See also

Kolmogorov extension theorem

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1. Theorem 2.2.3