In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Kolmogorov continuity theorem is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 that guarantees that a stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 that satisfies certain constraints on the moments

Moment (mathematics)

In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

 of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Andrey Nikolaevich Kolmogorov

Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

.

Statement of the theorem

Let be a stochastic process, and suppose that for all times , there exist positive constants such that


for all . Then there exists a continuous version of , i.e. a process such that

• is sample continuous
  Sample continuous process
  In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.-Definition:Let be a probability space. Let X : I × Ω → S be a stochastic process, where the index set I and state space S...
  
  ;
• for every time ,

Example

In the case of Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

on , the choice of constants , , will work in the Kolmogorov continuity theorem.