Kolmogorov continuity theorem explained

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

Let

(S,d) be some complete metric space, and let X\colon [ 0 , + inf ty) x \Omega \to S be a stochastic process. Suppose that for all times T > 0 , there exist positive constants \alpha, \beta, K such thatE [d(X _{t,} X ^{\alpha]} _{s)} \leq K | t - s | ^{1}

for all

0 \leq s, t \leq T . Then there exists a modification \tilde{X} of X that is a continuous process, i.e. a process \tilde{X}\colon [ 0 , + inf ty) x \Omega \to S such that\tilde{X} is sample-continuous;

• for every time

t \geq 0 , P (X _{t} = \tilde{X} _{t)} = 1 .

Furthermore, the paths of

\tilde{X} are locally \gamma -Hölder-continuous for every 0 <\gamma<\tfrac\beta\alpha .

Example

In the case of Brownian motion on

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

See also

• Kolmogorov extension theorem

References

• Book: Daniel W. Stroock, S. R. Srinivasa Varadhan

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