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 of the theorem

Let ${\displaystyle (S,d)}$  be some complete metric space such that d is ${\displaystyle {\mathcal {B}}(S)\otimes {\mathcal {B}}(S)}$  measurable, and let ${\displaystyle X:[0,+\infty )\times \Omega \to S}$  be a stochastic process. Suppose that for all times ${\displaystyle T>0}$ , there exist positive constants ${\displaystyle \alpha ,\beta ,K}$  such that

${\displaystyle \mathbb {E} [d(X_{t},X_{s})^{\alpha }]\leq K|t-s|^{1+\beta }}$ 

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

• ${\displaystyle {\tilde {X}}}$  is sample-continuous;
• for every time ${\displaystyle t\geq 0}$ , ${\displaystyle \mathbb {P} (X_{t}={\tilde {X}}_{t})=1.}$ 

Furthermore, the paths of ${\displaystyle {\tilde {X}}}$  are locally ${\displaystyle \gamma }$ -Hölder-continuous for every ${\displaystyle 0<\gamma <{\tfrac {\beta }{\alpha }}}$ .

Example

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

See also

Kolmogorov extension theorem

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1. Theorem 2.2.3