The Daniell-Kolmogorov theorem seen in Lecture 5 is a very useful tool since it provides existence results for stochastic processes. Nevertheless, this theorem does not say anything about the paths of this process. The following theorem, due to Kolmogorov, precises that, under mild conditions, we can work with processes whose paths are quite regular.
Definition. A function is said to be Hölder continuous with exponent if there exists a constant such that for ,
Hölder functions are of course in particular continuous.
Definition. A stochastic process is called a modification of the process if for every ,
We can observe that if is a modification of then has the same distribution as (because they need to have the same finite-dimensional distributions).
Theorem. (Kolmogorov continuity theorem) Let . If a -dimensional process defined on a probability space satisfies for ,
then there exists a modification of the process that is a continuous process and whose paths are -Hölder for every .
We make the proof for and let the reader extend it as an exercise to the case . For , we denote
Let . From Chebychev’s inequality:
Therefore, since , we deduce
From the Borel-Cantelli lemma, we can thus find a set such that and such that for , there exists such that for ,
In particular, there exists an almost surely finite random variable such that for every ,
We now claim that the paths of the restricted process are consequently -Hölder on . Indeed, let , . We can find such that
We now pick an increasing and stationary sequence converging toward , such that and
In the same way, we can find an analogue sequence that converges toward and such that and are neighbors in . We have then:
where the above sums are actually finite.
Hence the paths of are -Hölder on the set . For , let be the unique continuous function that agrees with on . For , we set . The process is the desired modification of .