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.

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

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

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

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

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

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

• Kolmogorov extension theorem

• Daniel W. Stroock, S. R. Srinivasa Varadhan (1997). Multidimensional Diffusion Processes. Springer, Berlin. ISBN 978-3-662-22201-0. p. 51

• Theorems about stochastic processes

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