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 ${\displaystyle X:[0,+\infty )\times \Omega \to \mathbb {R} ^{n}}$  be a stochastic process, and suppose that for all times ${\displaystyle T>0}$ , there exist positive constants ${\displaystyle \alpha ,\beta ,K}$  such that

${\displaystyle \mathbb {E} \left[|X_{t}-X_{s}|^{\alpha }\right]\leq K|t-s|^{1+\beta }}$ 

for all ${\displaystyle 0\leq s,t\leq T}$ . Then there exists a continuous version of ${\displaystyle X}$ , i.e. a process ${\displaystyle {\tilde {X}}:[0,+\infty )\times \Omega \to \mathbb {R} ^{n}}$  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 almost surely ${\displaystyle \gamma }$ -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.

References

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