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 constrains 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.

[edit] 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 constants $α,β, D > 0$ such that

$\mathbb{E} \left[ | X_{t} - X_{s} |^{\alpha} \right] \leq D | t - s |^{1 + \beta}$

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

[edit] Example

In the case of Brownian motion on $\mathbb{R}^{n}$ , the choice of constants $α = 4$ , $β = 1$ , $D = n (n + 2)$ will work in the Kolmogorov continuity theorem.