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 [math]\displaystyle{ (S,d) }[/math] be some complete metric space, and let [math]\displaystyle{ X\colon [0, + \infty) \times \Omega \to S }[/math] be a stochastic process. Suppose that for all times [math]\displaystyle{ T \gt 0 }[/math], there exist positive constants [math]\displaystyle{ \alpha, \beta, K }[/math] such that

[math]\displaystyle{ \mathbb{E} [d(X_t, X_s)^\alpha] \leq K | t - s |^{1 + \beta} }[/math]

for all [math]\displaystyle{ 0 \leq s, t \leq T }[/math]. Then there exists a modification [math]\displaystyle{ \tilde{X} }[/math] of [math]\displaystyle{ X }[/math] that is a continuous process, i.e. a process [math]\displaystyle{ \tilde{X}\colon [0, + \infty) \times \Omega \to S }[/math] such that

• [math]\displaystyle{ \tilde{X} }[/math] is sample-continuous;
• for every time [math]\displaystyle{ t \geq 0 }[/math], [math]\displaystyle{ \mathbb{P} (X_t = \tilde{X}_t) = 1. }[/math]

Furthermore, the paths of [math]\displaystyle{ \tilde{X} }[/math] are locally [math]\displaystyle{ \gamma }[/math]-Hölder-continuous for every [math]\displaystyle{ 0\lt \gamma\lt \tfrac\beta\alpha }[/math].

Example

In the case of Brownian motion on [math]\displaystyle{ \mathbb{R}^n }[/math], the choice of constants [math]\displaystyle{ \alpha = 4 }[/math], [math]\displaystyle{ \beta = 1 }[/math], [math]\displaystyle{ K = n (n + 2) }[/math] will work in the Kolmogorov continuity theorem. Moreover, for any positive integer [math]\displaystyle{ m }[/math], the constants [math]\displaystyle{ \alpha = 2m }[/math], [math]\displaystyle{ \beta = m-1 }[/math] will work, for some positive value of [math]\displaystyle{ K }[/math] that depends on [math]\displaystyle{ n }[/math] and [math]\displaystyle{ m }[/math].

See also

• Kolmogorov extension theorem

References

• Daniel W. Stroock, S. R. Srinivasa Varadhan (1997). Multidimensional Diffusion Processes. Springer, Berlin. ISBN 978-3-662-22201-0.  p. 51