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

$\mathbb{𝔼}[d(X_t,X_s{)}^{\alpha }]\le K|t-s{|}^{1+\beta }$

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

• $\tilde{X}$ is sample-continuous;
• for every time $t\ge 0$, $\mathbb{\mathbb{P}}(X_t={\tilde{X}}_t)=1.$

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

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

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