Kolmogorov continuity theorem

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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} \left[ d(X_{t}, X_{s})^{\alpha} \right] \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

Furthermore, the paths of \tilde{X} are almost surely \gamma-Hölder continuous 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