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

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

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

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

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

Example

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

See also

• Kolmogorov extension theorem

References

• Daniel W. Stroock, S. R. Srinivasa Varadhan (1997). Multidimensional Diffusion Processes. Springer, Berlin. ISBN 978-3-662-22201-0. p. 51