Kolmogorov continuity theorem


Kolmogorov continuity theorem

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.

tatement of the theorem

Let X : [0, + infty) imes Omega o mathbb{R}^{n} be a stochastic process, and suppose that for all times T > 0, there exist constants alpha, eta, D > 0 such that

:mathbb{E} left [ | X_{t} - X_{s} |^{alpha} ight] leq D | t - s |^{1 + eta}

for all 0 leq s, t leq T. Then there exists a continuous version of X, i.e. a process ilde{X} : [0, + infty) imes Omega o mathbb{R}^{n} such that

* ilde{X} is sample continuous;
* for every time t geq 0, mathbb{P} (X_{t} = ilde{X}_{t}) = 1.

Example

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

References

* Theorem 2.2.3


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Kolmogorov's theorem — is any of several different results by Andrey Kolmogorov:;In statistics * Kolmogorov Smirnov test;In probability theory * Hahn Kolmogorov theorem * Kolmogorov existence theorem * Kolmogorov continuity theorem * Kolmogorov s three series theorem * …   Wikipedia

  • Continuity theorem — In mathematics and statistics, the continuity theorem may refer to one of the following results: the Lévy continuity theorem on random variables; the Kolmogorov continuity theorem on stochastic processes. See also Continuity (disambiguation) …   Wikipedia

  • Continuity — may refer to: In mathematics: The opposing concept to discreteness; common examples include: Continuous probability distribution or random variable in probability and statistics Continuous game, a generalization of games used in game theory Law… …   Wikipedia

  • Andrey Kolmogorov — Infobox Scientist name = Andrey Kolmogorov birth date = birth date|1903|4|25 birth place = Tambov, Imperial Russia nationality = Russian death date = death date and age|1987|10|20|1903|4|25 death place = Moscow, USSR field = Mathematician work… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • Stochastic process — A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or deterministic system) in probability theory. Instead of dealing with only one possible reality of how the process might evolve under time (as is… …   Wikipedia

  • Liste de théorèmes — par ordre alphabétique. Pour l établissement de l ordre alphabétique, il a été convenu ce qui suit : Si le nom du théorème comprend des noms de mathématiciens ou de physiciens, on se base sur le premier nom propre cité. Si le nom du théorème …   Wikipédia en Français

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Per Martin-Löf — in 2004 Born May 8, 1942 (194 …   Wikipedia