Kolmogorov’s continuity theorem

Let $X={\{X_t\}}_{t\ge 0}$ be a process satisfying the following condition$:$ For all $T>0$ there exist positive constants $\alpha$, $\beta$, $D$ such that

$$E[{|X_t-X_s|}^{\alpha }]\le D{|t-s|}^{1+\beta }\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.25em}0}\le s,t\le T.$$

Then there exists a continuous modification of $X$.

Title	Kolmogorov’s continuity theorem
Canonical name	KolmogorovsContinuityTheorem
Date of creation	2013-03-22 15:43:43
Last modified on	2013-03-22 15:43:43
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	9
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 60G07
Related topic	DistributionsOfAStochasticProcess