Kolmogorov continuity theorem: Difference between revisions

Revision as of 19:53, 9 February 2016

Latest revision as of 20:59, 14 April 2025

(17 intermediate revisions by 16 users not shown)

Line 1:

{{Short description|Mathematical theorem}}

In [[mathematics]], the '''Kolmogorov continuity theorem''' is a [[theorem]] that guarantees that a [[stochastic process]] that satisfies certain constraints on the [[moment (mathematics)|moments]] of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the [[Soviet Union|Soviet]] [[mathematician]] [[Andrey Kolmogorov|Andrey Nikolaevich Kolmogorov]].

==Statement of the theorem==

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

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

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

* <math>\tilde{X}</math> is [[sample-continuous process|sample-continuous]];

* for every time <math>t \geq 0</math>, <math>\mathbb{P} (X_t = \tilde{X}_t) = 1.</math>

Furthermore, the paths of <math>\tilde{X}</math> are almost surelylocally [[Hölder continuity|<math>\gamma</math>-Hölder-continuous]] for every <math>0<\gamma<\tfrac\beta\alpha</math>.

==Example==

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

==See also==

* [[Kolmogorov extension theorem]]

==References==

* {{cite book | author=Øksendal Daniel W. Stroock, BerntS. KR. Srinivasa Varadhan | authorlink=BerntDaniel ØksendalW. |Stroock, title=StochasticS. DifferentialR. Equations:Srinivasa AnVaradhan Introduction| withtitle=Multidimensional Diffusion ApplicationsProcesses | publisher=Springer, Berlin | year=20031997 | isbn=978-3-540662-0475822201-10}} Theorem 2p.2.3&nbsp;51

[[Category:StochasticTheorems about stochastic processes]]

[[Category:Statistical theorems]]

[[Category:Probability theorems]]