# The Kolmogorov Continuity Theorem

One of the common themes throughout the theory of continuous-time stochastic processes, is the importance of choosing good versions of processes. Specifying the finite distributions of a process is not sufficient to determine its sample paths so, if a continuous modification exists, then it makes sense to work with that. A relatively straightforward criterion ensuring the existence of a continuous version is provided by Kolmogorov’s continuity theorem.

For any positive real number ${\gamma}$, a map ${f\colon E\rightarrow F}$ between metric spaces E and F is said to be ${\gamma}$-Hölder continuous if there exists a positive constant C satisfying

 $\displaystyle d(f(x),f(y))\le Cd(x,y)^\gamma$

for all ${x,y\in E}$. The smallest value of C satisfying this inequality is known as the ${\gamma}$-Hölder coefficient of ${f}$. Hölder continuous functions are always continuous and, at least on bounded spaces, is a stronger property for larger values of the coefficient ${\gamma}$. So, if E is a bounded metric space and ${\alpha\le\beta}$, then every ${\beta}$-Hölder continuous map from E is also ${\alpha}$-Hölder continuous. In particular, 1-Hölder and Lipschitz continuity are equivalent.

Kolmogorov’s theorem gives simple conditions on the pairwise distributions of a process which guarantee the existence of a continuous modification but, also, states that the sample paths ${t\mapsto X_t}$ are almost surely locally Hölder continuous. That is, they are almost surely Hölder continuous on every bounded interval. To start with, we look at real-valued processes. Throughout this post, we work with repect to a probability space ${(\Omega,\mathcal F, {\mathbb P})}$. There is no need to assume the existence of any filtration, since they play no part in the results here

Theorem 1 (Kolmogorov) Let ${\{X_t\}_{t\ge0}}$ be a real-valued stochastic process such that there exists positive constants ${\alpha,\beta,C}$ satisfying

 $\displaystyle {\mathbb E}\left[\lvert X_t-X_s\rvert^\alpha\right]\le C\lvert t-s\vert^{1+\beta},$

for all ${s,t\ge0}$. Then, X has a continuous modification which, with probability one, is locally ${\gamma}$-Hölder continuous for all ${0 < \gamma < \beta/\alpha}$.