Kolmogorov continuity criteria

Contents

The Kolmogorov continuity criteria is a simple moment condition that ensures the existence of a Hölder-continuous version of a given process in a complete metric space. It has many applications in Probability Theory. For example, one can show the Brownian Motion is almost surely locally Hölder continuous of order

p

, for every

p \in(0, \frac{1}{2})

Theorem (The Kolmogorov continuity criteria)
Let

X

be a process on

\mathbb{R}^d

with values in a complete metric space

(S, \, \rho)

such that

\begin{equation}E \left\{ \rho \left(X_s, X_t\right) \right\}^a \leq c|s-t|^{d+b}, \quad s, t \in \mathbb{R}^d,\end{equation}

for some constants

a, b > 0

. Then there exists a modification

\tilde{X}

X

that is locally Hölder continuous of order

p

, for every

p \in(0, \frac{b}{a})

, i.e. process

\tilde{X} \colon [0, \infty)~\times~\Omega

such that

\tilde{X}

is locally Hölder continuous and

\begin{equation*}\forall t \geq 0, \quad \mathbb{P} (\tilde{X}_t = X_t) = 1\end{equation*}

Motivation

The fundamental motivation for Kolmogorov's continuity criterion is to provide a practical way to prove that random processes have continuous sample paths. Here's why this is important:

Problem formulation

Direct Verification Challenge

When working with stochastic processes, directly verifying continuity of sample paths can be extremely difficult because:

We need to check continuity at every point
Random processes often have complex dependencies
Traditional continuity arguments may not work well with randomness

Moment-Based Alternative

Kolmogorov's insight was that instead of checking continuity directly, we could look at the moments of increments:For a process

X_t

, if we can show that for some

p > 0

and

\alpha > 0

\begin{equation*}E \left\{ \rho \left(X_s, X_t\right) \right\}^a \leq c|s-t|^{d+b}, \quad s, t \in \mathbb{R}^d,\end{equation*}

Then under mild additional conditions, the process has a continuous modification.

Practical Applications for Brownian Motion

This criterion is particularly valuable for Brownian Motion

B(t)

. We have:

\begin{equation*}E[|B(t) - B(s)|^4] = 3|t-s|^2\end{equation*}

Taking

p=4

and

\alpha=1

, the criterion immediately gives continuity.

Proof

Construction approximation sequence

We can consider the restriction of

X

[0,1]^d

without the loss of generality. We can define the approximation dataset

G_n

of points

[0, 1]^d

such that the coordinates of

2^n x

are positive integers and

| G_n | = (2^n + 1)^d

. The for each

u \in [0, 1]^d

we choose

\pi_n (u) \in G_n

as close to

u

as possible, so that

| \pi_n (u) - u | \leq 2^{-n}

and

|\pi_n (u) - \pi_{n-1} (u)| \leq | \pi_n (u) - u | + | u - \pi_{n-1} (u) | \leq 3 2^{-n}

continuityKolmogorov approximation_set.png — The sets $G_3$ and $G_4$ in $[0, 1]^2$

Then consider the random variable

\begin{equation*}Y_n = \max \left\{ \rho(X_s, X_t); \quad |s-t| \leq \frac{3}{2^n} \right\}\end{equation*}

and since

G_{n-1} \subset G_n

, which means

\forall u \in [0,1]^d \colon \pi_{n-1} (u), \pi_{n} (u) \in G_n

. So we get

\forall u \in [0,1]^d

\begin{equation*}\rho(X_{\pi_{n}(u)}, X_{\pi_{n-1}(u)}) \leq Y_n\end{equation*}

For the set

G = \bigcup_{n\geq0} G_n

we claim that

\begin{equation}\sup_{s, t \in G; \newline |s-t| \leq 2^{-k}} \rho(X_s, X_t) \leq 3 \sum_{n\geq k} Y_n\end{equation}

To prove this, consider

n \geq k

such that

s, t \in G_n

, so that

s = \pi_n (s)

and

t = \pi_n (t)

. Assuming

| s - t| \leq 2^{-k}

, we have

\begin{equation*}|\pi_k (s) - \pi_{k} (t)| \leq | \pi_k (u) - s | + |s - t| + | t - \pi_k (t) | \leq 3 2^{-k}\end{equation*}

and thus

\begin{equation*}\rho(X_{\pi_{k}(s)}, X_{\pi_k(t)}) \leq Y_k\end{equation*}

Next, for

u \in \{ s, t \}

\begin{equation*}\rho(X_u, X_{\pi_k(u)}) = \rho(X_{\pi_{n}(u)}, X_{\pi_k(u)}) = \sum_{k \leq l \leq n} \rho(X_{\pi_{l+1}(u)}, X_{\pi_l(u)}),\end{equation*}

and since

\rho(X_{\pi_{l+1}(u)}, X_{\pi_l(u)}) \leq Y_{l+1}

, we obtain

\begin{equation*}\rho(X_u, X_{\pi_k(u)}) \leq \sum_{ l \geq k} Y_{l+1}\end{equation*}

We then use the previous inequalities and get

\begin{equation*}\rho(X_s, X_t) \leq \rho(X_s, X_{\pi_k(s)}) + \rho(X_{\pi_k(s)}, X_{\pi_k(t)}) + \rho(X_{\pi_k(t)}, X_t) \leq 3 \sum_{ l \geq k} Y_{l+1}\end{equation*}

The inequality for maximum

The number of points on which we consider the maximum process

Y_n

can be evaluated as follow

\begin{equation*}\left| \left\{(s,t) \in G_n \colon |s-t| \leq \frac{3}{2^n} \right\} \right| \leq 3^d 2^{nd}\end{equation*}

Then using Chebyshev inequality and condition (1):

\begin{align*}P \{ Y_n > \frac{1}{2^{pn}}\} &= P \left\{ \bigcup_{(s,t) \colon |s-t| \leq 3 2^{-n}} \rho(X_s, X_t) > \frac{1}{2^{pn}} \right\} \leq \sum_{(s,t) \colon |s-t| \leq 3 2^{-n} } P \left\{ \rho(X_s, X_t) > \frac{1}{2^{pn}}\right\} \leq \\&\leq \sum_{(s,t) \colon |s-t| \leq 3 2^{-n} } P \left\{ \rho(X_s, X_t)^a > \frac{1}{2^{pna}}\right\} \leq C(d) 2^{n(pa - b)}\end{align*}

where the constant

C(d)

depends only on

d

Now we can verify that

X

If we choose

p \in (0, \frac{b}{a})

\begin{equation*}\sum_{n=1}^{\infty} P \left\{ Y_n \geq \frac{1}{2^{pn}} \right\} < \infty\end{equation*}

By Borel-Cantelli Lemma, we get that

\exists N_0 \, \forall n \geq N_0 \colon Y_n \leq 2^{-p n}

a.s. This implies convergence of the series

\sum_{n\geq0} Y_n

a.s.

Hölder continuous of $X$

Thus

\begin{equation}\sup \left\{ \rho(X_s, X_t) \colon s, t \in G, |s-t| \leq 2^{-m} \right\} \leq 3 \sum_{n\geq m} Y_n \leq 3 \sum_{n\geq m} 2^{-pn} \leq \frac{2^{-pm}}{1 - 2^{-p}}\end{equation}

showing that

X

is a.s. Hölder continuous on

G

of order

p

.
In particular,

X

agrees a.s. on

\bigcup_n \mathcal{D}_n

with a continuous process

\tilde{X}

[0,1]^d

, and we note that the Hölder continuity of

\tilde{X}

G

extends with the same order

p

to the entire cube

[0,1]^d

.
To show that

\tilde{X}

is a version of

X

, fix any

t \in[0,1]^d

, and choose

t_1, t_2, \ldots \in G

with

t_n \rightarrow t

. Then

X_{t_n}=\tilde{X}_{t_n}

a.s. for each

n

. Since also

X_{t_n} \stackrel{P}{\rightarrow} X_t

by (1) and

\tilde{X}_{t_n} \rightarrow \tilde{X}_t

a.s. by continuity, we get

X_t=\tilde{X}_t

a.s.

Kolmogorov continuity criteria

Motivation

Motivation

Direct Verification Challenge

Moment-Based Alternative

Practical Applications for Brownian Motion

Proof

Proof

Construction approximation sequence

The inequality for maximum

Hölder continuous of $X$

References

References

Kolmogorov continuity criteria

Motivation

Proof

Motivation

Motivation

Direct Verification Challenge

Moment-Based Alternative

Practical Applications for Brownian Motion

Proof

Proof

Construction approximation sequence

The inequality for maximum

Hölder continuous of XXX

References

References

Hölder continuous of $X$