From the Kolmogorov continuity theorem to Brownian motions

Kolmogorov continuity theorem

In stochastic analysis, the Kolmogorov continuity thoerem guarantees the Hölder continuity of a stochastic process using a distance inequality.

Kolmogorov continuity theorem

Let $(S,d)$ be some metric space, and $X:[0,T] \rightarrow S$ be a stochastic process. Suppose there exist positive constants $\alpha,\beta,K$ such that

$\mathbb E[d(X_t,X_s)^\alpha] \le K|t-s|^{1+\beta}$

for all $0\le s, t \le T$ , then there exists a modification $\tilde X$ of $X$ that is a continuous process, i.e., a process $\tilde X:[0,T]\rightarrow S$ such that

$\tilde X$ is sample continuous;

For every time $t \in [0,T]$ , $\mathbb P(X_t = \tilde X_t) = 1$ .

Furthermore, the paths of $\tilde X$ are locally $\gamma$ -Hölder continuous for every $0<\gamma<\frac{\beta}{\alpha}$ .

This result can be employed to obtain the Hölder continuity of a large variety of stochastic processes, and we begin with the simplest example, the standard Brownian motion.

Samples of the trajectories of the Brownian motion.

The Brownian motion $\{B_t\}_{t\ge 0}$ is a Markov process such that for any $t>0$ , the random variable $B_t$ lives in the Gaussian distribution $\mathcal N(0,t)$ . Therefore, we deduce

$\mathbb E|B_t - B_s|^2 = |t-s|$

for any $s,t \ge 0$ . It is important to note that the property above is NOT enough to deduce the Hölder continuity of $B_t$ . One may consider to divide the LHS by $|t-s|$ and take the supremum to obtain

$\displaystyle\mathbb E \bigg[\sup_{s,t\ge 0} \frac{|B_t - B_s|^2}{t-s} \bigg]\le 1$ ,

but this is absolutely incorrect because it changes the order of the expectation and the supremum.

The correct way to prove the continuity is to observe that for any positive integer $m$ , there exists a constant $C$ depending on $m$ such that

$\mathbb E|B_t - B_s|^{2m} \le C|t-s|^m$

for any $s,t \ge 0$ . This is because $B_t - B_s$ is the Gaussian distribution with variance $|t-s|$ . Then applying the Kolmogorov continuity theorem, we deduce that the Brownian motion $B_t$ is $\gamma$ -Hölder continuous for any

$\displaystyle 0 < \gamma < \frac{m-1}{2m}$ .

Finally, since $m$ can be sufficiently large, $\gamma$ can be arbitrarily close to $\frac12$ .

Karhunen-Loève expansion

The Brownian motion has an alternative expression, the Karhunen-Loève expansion. The Brownian motion $B_t$ can be written as the sum of an infinite series,

$\displaystyle B_t = \sum_{n=1}^\infty \xi_n \frac{\sin\big((n-\frac12)t\big)}{n-\frac12}$ ,

where $\{\xi_n\}_{n=1}^\infty$ are independent Gaussian random variables in $\mathcal N(0,1)$ . We claim that we can also derive the Hölder continuity of $B_t$ from the Karhunen-Loève expansion. In fact, we have the following general result.

Consider the stochastic process $X_t:[0,+\infty) \rightarrow \mathbb R$ given by

$\displaystyle X_t = \sum_{n=1}^\infty \xi_n c_n(t)$ ,

where $\{\xi_n\}_{n=1}^\infty$ are independent Gaussian random variables in $\mathcal N(0,1)$ , and $\{c_n(t)\}_{n=1}^\infty$ are a set of functions satisfying

$\displaystyle |c_n(t) - c_n(s)| \le |t - s|$

$\displaystyle |c_n(t)| \le \frac1n$

for any $s,t\ge 0$ and $n\in\mathbb N$ , then for any positive integer $m$ , there exists a constant $m$ such that

$\displaystyle \mathbb E|X_t - X_s|^{2m} \le C |t-s|^m$

As a consequence, $X_t$ is $\gamma$ -Hölder continuous for any $0<\gamma<\frac12$ .

Proof By direct calculation, we have

$\displaystyle \mathbb E|X_t - X_s|^{2m} = \mathbb E \bigg( \sum_{n=1}^\infty \xi_n \big(c_n(t) - c_n(s)\big) \bigg)^{2m} = \mathbb E \bigg( \sum_{n=1}^\infty \xi_n^2 \big(c_n(t) - c_n(s)\big)^2 \bigg)^m$

Here, the odd power of $\xi_n$ does not contribute to the expectation. Next, we expand the RHS with $m$ indices $n_1,\cdots,n_m\in\mathbb N$ to obtain

$\displaystyle \begin{aligned} \mathbb E|X_t - X_s|^{2m} & = \sum_{n_1 = 1}^\infty \cdots \sum_{n_m = 1}^\infty \mathbb E\big(\xi_{n_1}^2 \cdots \xi_{n_m}^2\big) \big(c_{n_1}(t) - c_{n_1}(s)\big)^2 \cdots \big(c_{n_m}(t) - c_{n_m}(s)\big)^2 \\ & \le C \sum_{n_1 = 1}^\infty \cdots \sum_{n_m = 1}^\infty \mathbb E\xi_{n_1}^2 \cdots \mathbb E\xi_{n_m}^2 \big(c_{n_1}(t) - c_{n_1}(s)\big)^2 \cdots \big(c_{n_m}(t) - c_{n_m}(s)\big)^2 \\ & = C \bigg(\sum_{n=1}^\infty \big(c_{n}(t) - c_{n}(s)\big)^2\bigg)^m. \end{aligned}$

Here, $\xi_{n_1},\cdots,\xi_{n_m}$ (may have duplicated ones) are all Gaussian random variables, and $C$ is a constant depending on $m$ . Now we only need to verify the inequality

$\displaystyle \sum_{n=1}^\infty \big(c_n(t) - c_n(s)\big)^2 \le C|t-s|$

To prove the inequality, divide the summation in the LHS into two parts.

$\displaystyle \begin{aligned} \sum_{n=1}^\infty \big(c_n(t) - c_n(s)\big)^2 & = \sum_{n|t-s|\le 1} \big(c_n(t) - c_n(s)\big)^2 + \sum_{n|t-s|> 1} \big(c_n(t) - c_n(s)\big)^2 \\ & = \sum_{n|t-s|\le 1} |t-s|^2 + \sum_{n|t-s|> 1} \frac{4}{n^2} \\ & \le |t-s| + 5|t-s| \\ & = 6|t-s|, \end{aligned}$

and thus we obtain the desired result.

Summary

We employ the Kolmogorov continuity theorem to obtain the Hölder continuity of the Brownian motion and a general class of stochastic processes. Readers can also refer to Lemma 3.2 of my recent paper for an interesting application of the result.

Xuda Ye