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
[edit]Let be some complete metric space, and let be a stochastic process. Suppose that for all times , there exist positive constants such that
for all . Then there exists a modification of that is a continuous process, i.e. a process such that
- is sample-continuous;
- for every time ,
Furthermore, the paths of are locally -Hölder-continuous for every .
Example
[edit]In the case of Brownian motion on , the choice of constants , , will work in the Kolmogorov continuity theorem. Moreover, for any positive integer , the constants , will work, for some positive value of that depends on and .
See also
[edit]References
[edit]- Daniel W. Stroock, S. R. Srinivasa Varadhan (1997). Multidimensional Diffusion Processes. Springer, Berlin. ISBN 978-3-662-22201-0. p. 51