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
Let $$(S,d)$$ be some complete metric space, and let $$X\colon [0, + \infty) \times \Omega \to S$$ 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 $$\tilde{X}$$ of $$X$$ that is a continuous process, i.e. a process $$\tilde{X}\colon [0, + \infty) \times \Omega \to S$$ 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 locally $\gamma$-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. Moreover, for any positive integer $$m$$, the constants $$\alpha = 2m$$, $$\beta = m-1$$ will work, for some positive value of $$K$$ that depends on $$n$$ and $$m$$.