Kolmogorov-Chentsov theorem

The set of Kolmogorov-Tschenzow even continuity theorem of Kolmogorov-Tschenzow called, is a mathematical theorem of probability theory and deals with properties of paths or realizations of stochastic processes . It makes a statement about when modifications of a stochastic process are continuous or locally wood- continuous . The statement goes back in a simpler form to Andrei Nikolajewitsch Kolmogorow and was generalized accordingly by Nikolai Nikolajewitsch Tschenzow in 1956.

The theorem is used, for example, in the construction of the Wiener process , where it guarantees the existence of continuous paths.

statement

A real-valued stochastic process is given , which has the non-negative real numbers as an index set. So it is . Furthermore, let there be real numbers for every such that ${\displaystyle X=(X_{t})_{t\in I}}$ ${\displaystyle I=[0,\infty )}$ ${\displaystyle T\in I}$ ${\displaystyle a,b,C>0}$

${\displaystyle \operatorname {E} (|X_{t}-X_{s}|^{a})\leq C|t-s|^{1+b}}$

applies to all of the interval . ${\displaystyle s,t}$ ${\displaystyle [0,T]}$

Then there is a modification of which has locally holier-continuous paths of order for all . ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle h}$ ${\displaystyle h\in (0,{\tfrac {b}{a}})}$

In addition, there is then a finite number for each such that ${\displaystyle \varepsilon >0}$ ${\displaystyle K=K(\varepsilon ,T,a,b,C,h)}$

${\displaystyle P\left(|X_{t}^{*}-X_{s}^{*}|\leq K|t-s|^{h}{\text{ für }}s,t\in [0,T]\right)\geq 1-\varepsilon }$.

Example: Vienna Trial

The Wiener process is a real-valued process with an index set , which is characterized by the following properties: ${\displaystyle W=(W_{t})_{t\in I}}$ ${\displaystyle I=[0,\infty )}$

1. ${\displaystyle W_{0}=0}$.
2. ${\displaystyle W}$is a process with independent gains .
3. ${\displaystyle W}$is a process with steady growth .
4. The increases are normally distributed , so it applies . ${\displaystyle W_{t}\sim {\mathcal {N}}_{0,t}}$
5. The paths of the process are almost certainly steady.

With Kolmogorov-Tschenzow's theorem one can now show that the fifth condition is redundant, i.e. H. if the first four conditions apply to a process, there is always a modification of the process which fulfills the fifth condition.

Because due to the stationary independent increases and the scaling properties of the normal distribution applies

${\displaystyle W_{t}-W_{s}\sim {\sqrt {t-s}}\;W_{1}\sim {\mathcal {N}}_{0,t-s}}$.

With the calculation rules of the expected value it follows

${\displaystyle \operatorname {E} \left((W_{t}-W_{s})^{2n}\right)=\operatorname {E} \left(({\sqrt {t-s}}\;W_{1})^{2n}\right)=\operatorname {E} (W_{1}^{2n})|t-s|^{n}}$

and for example through the torque generating function one obtains . According to Kolmogorow-Tschenzow's theorem with and, there is now a locally Hölder- continuous modification of the process for each and every one . ${\displaystyle \operatorname {E} (W_{1}^{2n})={\tfrac {(2n)!}{2^{n}n!}}}$ ${\displaystyle a=2n,b=n-1}$ ${\displaystyle C=C(n)={\tfrac {(2n)!}{2^{n}n!}}}$ ${\displaystyle h\in (0,{\tfrac {n-1}{2n}})}$ ${\displaystyle n\geq 2}$ ${\displaystyle h}$ ${\displaystyle W}$

Generalizations

The statement of the sentence also applies without further restrictions to processes that take on values ​​in Polish areas . With changes in the amount of time, however, you have to make stronger demands.

Individual evidence

1. ↑ Nikolai Nikolaevich Chentsov (Obituary) Limited online access to Chentsov's obituary in 1993 Russ. Math. Surv. 48 161 with an overview of his life's work. Retrieved November 14, 2015.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 470-473 , doi : 10.1007 / 978-3-642-36018-3 .