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 .
*

**Proof:**

We make the proof for and let the reader extend it as an exercise to the case . For , we denote

and

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.

Therefore,

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 .

The theorem is stated in the form that “suppose for a fixed interval [0,T] there exist a set of parameters \alpha, \epsilon, c such that (some inequality holds), then there exists a continuous modification of the process on [0,T]”. The set of parameters depend on the interval.

We do not hope for a set of parameters such that the condition inequality holds on [0,\infty) globally, because the concept of Holder continuity is more often discussed only locally. It would restrict ourselves too much to discuss about “globally Holder”. Is this correct?

Yes, indeed, most of the processes, including the Brownian motion, are not globally Holder but only locally: The Holder constant depends on the interval.

Thanks Professor.

Just remember

(i) D

(ii) the set \Omega^*, which has full measure

(iii) paths of X restricted on \Omega^* are \gamma-Holder on D

it shouldn’t be difficult to memorize the theorem.

Is the assertion |s-t| <= 2^{-n} backwards? We find |X_t-X_s| <= 2^{-alpha n} which we want to be <= |s-t|^{\alpha}, for which 2^{-n} <= |s-t| is more useful.

The proof was fine, because implies that is -Holder, but I slightly edited the post to make it more clear. Thanks for the comment.