Lecture 6. The Kolmogorov continuity theorem

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 $f:\mathbb{R}_{\ge 0} \rightarrow \mathbb{R}^d$ is said to be Hölder continuous with exponent $\alpha >0$ if there exists a constant $C >0$ such that for $s,t \in \mathbb{R}_{\ge 0}$ ,

$\| f(t)-f(s) \| \le C \mid t-s \mid^{\alpha}.$

Hölder functions are of course in particular continuous.

Definition. A stochastic process $(\tilde{X}_t)_{t \geq 0}$ is called a modification of the process $(X_t)_{t \geq 0}$ if for every $t \geq 0$ , $\mathbb{P} \left( X_t = \tilde{X}_t \right)=1.$

We can observe that if $(\tilde{X}_t)_{t \geq 0}$ is a modification of $(X_t)_{t \geq 0}$ then $(\tilde{X}_t)_{t \geq 0}$ has the same distribution as $(X_t)_{t \geq 0}$ (because they need to have the same finite-dimensional distributions).

Theorem. (Kolmogorov continuity theorem) Let $\alpha, \varepsilon, c >0$ . If a $d$ -dimensional process $(X_t)_{t\in [0,1]}$ defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ satisfies for $s,t \in [0,1]$ ,

$\mathbb{E} \left( \| X_t - X_s \|^{\alpha} \right) \leq c \mid t-s \mid^{1+\varepsilon},$

then there exists a modification of the process $(X_t)_{t \in [0,1]}$ that is a continuous process and whose paths are $\gamma$ -Hölder for every $\gamma \in [0, \frac{\varepsilon}{\alpha} )$ .

Proof:

We make the proof for $d=1$ and let the reader extend it as an exercise to the case $d \ge 2$ . For $n \in \mathbb{N}$ , we denote

$\mathcal{D}_n=\left\{ \frac{k}{2^n}, k=0,...,2^{n} \right\}$

and

$\mathcal{D}=\cup_{n \in \mathbb{N}} \mathcal{D}_n.$

Let $\gamma \in [0, \frac{\varepsilon}{\alpha} )$ . From Chebychev’s inequality:

$\mathbb{P} \left( \max_{1 \le k \le 2^n} | X_{\frac{k}{2^n}} -X_{\frac{k-1}{2^n}} | \ge 2^{-\gamma n}\right)$
$=\mathbb{P} \left( \cup_{1 \le k \le 2^n} | X_{\frac{k}{2^n}} -X_{\frac{k-1}{2^n}} | \ge 2^{-\gamma n}\right)$
$\le \sum_{k=1}^{2^n} \mathbb{P} \left( | X_{\frac{k}{2^n}} -X_{\frac{k-1}{2^n}} | \ge 2^{-\gamma n}\right)$
$\le \sum_{k=1}^{2^n} \frac{\mathbb{E}\left( | X_{\frac{k}{2^n}} -X_{\frac{k-1}{2^n}} |^{\alpha}\right)}{2^{-\gamma \alpha n}}$
$\le c 2^{-n(\varepsilon-\gamma \alpha)}$

Therefore, since $\gamma \alpha < \varepsilon$ , we deduce

$\sum_{n=1}^{+\infty} \mathbb{P} \left( \max_{1 \le k \le 2^n} | X_{\frac{k}{2^n}} -X_{\frac{k-1}{2^n}} | \ge 2^{-\gamma n}\right)<+\infty.$

From the Borel-Cantelli lemma, we can thus find a set $\Omega^* \in \mathcal{F}$ such that $\mathbb{P} ( \Omega^*)=1$ and such that for $\omega \in \Omega^*$ , there exists $N(\omega)$ such that for $n \ge N(\omega)$ ,

$\max_{1 \le k \le 2^n} | X_{\frac{k}{2^n}} (\omega) -X_{\frac{k-1}{2^n}} (\omega)| \le 2^{-\gamma n}.$

In particular, there exists an almost surely finite random variable $C$ such that for every $n \ge 0$ ,

$\max_{1 \le k \le 2^n} | X_{\frac{k}{2^n}} (\omega) -X_{\frac{k-1}{2^n}} (\omega)| \le C 2^{-\gamma n}$

We now claim that the paths of the restricted process $X_{/\Omega^*}$ are consequently $\gamma$ -Hölder on $\mathcal{D}$ . Indeed, let $s,t \in \mathcal{D}$ , $t \neq s$ . We can find $n \ge 0$ such that

$\frac{1}{2^{n+1}} \le \mid s-t \mid \le \frac{1}{2^n}.$

We now pick an increasing and stationary sequence $(s_k)_{k \ge n}$ converging toward $s$ , such that $s_k \in \mathcal{D}_k$ and

$\mid s_{k+1}-s_k \mid =2^{-(k+1)} \quad \text{or} \quad 0.$

In the same way, we can find an analogue sequence $(t_k)_{k \ge n}$ that converges toward $t$ and such that $s_n$ and $t_n$ are neighbors in $\mathcal{D}_n$ . We have then:

$X_t - X_s=\sum_{i=n}^{+\infty}(X_{s_{i+1}} -X_{s_{i}}) +(X_{s_n}-X_{t_n})+\sum_{i=n}^{+\infty}(X_{t_{i}} -X_{t_{i+1}}),$

where the above sums are actually finite.

Therefore,
$| X_t - X_s |$
$\le C 2^{-\gamma n}+ 2 \sum_{k=n}^{+\infty} C 2^{-\gamma(k+1)}$
$\le 2C \sum_{k=n}^{+\infty} 2^{- \gamma k}$
$\le \frac{2C}{1-2^{-\gamma}} 2^{-\gamma n}$

Hence the paths of $X_{/\Omega^*}$ are $\gamma$ -Hölder on the set $\mathcal{D}$ . For $\omega \in \Omega^*$ , let $t\rightarrow \tilde{X}_t (\omega)$ be the unique continuous function that agrees with $t\rightarrow X_t (\omega)$ on $\mathcal{D}$ . For $\omega \notin \Omega^*$ , we set $\tilde{X}_t (\omega)=0$ . The process $(\tilde{X}_t)_{t \in [0,1]}$ is the desired modification of $(X_t)_{t \in [0,1]}$ . $\square$

5 Responses to Lecture 6. The Kolmogorov continuity theorem

zihui says:

February 28, 2014 at 9:10 pm

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?

- Fabrice Baudoin says:
  
  February 28, 2014 at 9:13 pm
  
  Yes, indeed, most of the processes, including the Brownian motion, are not globally Holder but only locally: The Holder constant depends on the interval.
  
zihui says:

February 28, 2014 at 9:36 pm

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.

etfoxall says:

July 18, 2014 at 4:39 pm

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.

- Fabrice Baudoin says:
  
  July 18, 2014 at 6:16 pm
  
  The proof was fine, because $|t-s| \le \frac{1}{2^n} \Rightarrow |X_t -X_s| \le C \frac{1}{2^{n\gamma}}$ implies that $X$ is $\gamma$ -Holder, but I slightly edited the post to make it more clear. Thanks for the comment.