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 $t \geq 0$ , $\mathbb{P} (X_{t} = \tilde{X}_{t}) = 1.$

Furthermore, the paths of $\tilde{X}$ are almost surely $\gamma$ for every $0<\gamma<\tfrac\beta\alpha$ .

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