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