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[edit]

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

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

Example[edit]

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[edit]