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 $(S,d)$ be some metric space, and let $X : [0, + \infty) \times \Omega \to S$ be a stochastic process. Suppose that for all times $T > 0$, there exist positive constants $\alpha, \beta, K$ such that

$\mathbb{E} [d(X_t, X_s)^\alpha] \leq K | t - s |^{1 + \beta}$

for all $0 \leq s, t \leq T$. Then there exists a modification of $X$ that is a continuous process, i.e. a process $\tilde{X} : [0, + \infty) \times \Omega \to S$ 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.

See also

Kolmogorov extension theorem

References

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