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 , a map between metric spaces E and F is said to be -Hölder continuous if there exists a positive constant C satisfying
for all . The smallest value of C satisfying this inequality is known as the -Hölder coefficient of . Hölder continuous functions are always continuous and, at least on bounded spaces, is a stronger property for larger values of the coefficient . So, if E is a bounded metric space and , then every -Hölder continuous map from E is also -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 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 . There is no need to assume the existence of any filtration, since they play no part in the results here
Theorem 1 (Kolmogorov) Let be a real-valued stochastic process such that there exists positive constants satisfying
for all . Then, X has a continuous modification which, with probability one, is locally -Hölder continuous for all .