# Kolmogorov’s continuity theorem

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

 $E[|X_{t}-X_{s}|^{\alpha}]\leq D|t-s|^{1+\beta}\,\,\ 0\leq s,t\leq T.$

Then there exists a continuous modification of $X$.

Title Kolmogorov’s continuity theorem KolmogorovsContinuityTheorem 2013-03-22 15:43:43 2013-03-22 15:43:43 georgiosl (7242) georgiosl (7242) 9 georgiosl (7242) Theorem msc 60G07 DistributionsOfAStochasticProcess