0.1 Kolmogorov’s Continuity Lemma
Let \(\{ X_{t},t\geq 0 \}\) be a stochastic process such that for all \(T^*>0\) there exist \(\alpha, \beta, C>0\) such that \[ \mathbb{E}[\lvert X_{t+h}-X_{t} \rvert^\alpha ]\leq Ch^{1+\beta}, \] for \(h>0\) and \(0 < t<T^*-h\). Then there exists a continuous version of \(X\).
The proof can be found in Chapter 8 of [[probability-theory-and-examples-durrett-pdf]].