Kolmogorov continuity theorem

In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.
Statement of the theorem

Let \( X : [0, + \infty) \times \Omega \to \mathbb{R}^{n} \) be a stochastic process, and suppose that for all times T > 0, there exist positive constants \( \alpha, \beta, K \) such that

\( \mathbb{E} \left[ | X_{t} - X_{s} |^{\alpha} \right] \leq K | t - s |^{1 + \beta} \)

for all \( 0 \leq s, t \leq T. Then there exists a continuous version of X, i.e. a process \( \tilde{X} : [0, + \infty) \times \Omega \to \mathbb{R}^{n} \) such that

\( \tilde{X} \) is sample continuous;
for every time \( t \geq 0, \mathbb{P} (X_{t} = \tilde{X}_{t}) = 1. \)

Example

In the case of Brownian motion on \( \mathbb{R}^{n} \), the choice of constants \( \alpha = 4, \beta = 1, K = n (n + 2) \) will work in the Kolmogorov continuity theorem.
References

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1. Theorem 2.2.3

Mathematics Encyclopedia