The Kolmogorov continuity theorem, Hölder continuity, and the Kolmogorov-Chentsov theorem

1 Modifications

Let $(\mathrm{\Omega },ℱ,P)$ be a probability space, let $I$ be a nonempty set, and let $(E,ℰ)$ be a measurable space. A stochastic process with index set $I$ and state space $E$ is a family ${(X_t)}_{t\in I}$ of random variables $X_t:(\mathrm{\Omega },ℱ)\to (E,ℰ)$. If $X$ and $Y$ are stochastic processes, we say that $X$ is a modification of $Y$ if for each $t\in I$,

$$P\{\omega \in \mathrm{\Omega }:X_t(\omega )=Y_t(\omega )\}=1.$$

Lemma 1.

If $X$ is a modification of $Y$, then $X$ and $Y$ have the same finite-dimensional distributions.

Proof.

For $t_1,\mathrm{\dots },t_n\in I$, let $A_i\in ℰ$ for each $1\le i\le n$, and let

$$A=\bigcap _{i=1}^n{X}_{t_i}^{-1}(A_i)\in ℱ,B=\bigcap _{i=1}^n{Y}_{t_i}^{-1}(A_i)\in ℱ.$$

If $\omega \in A\setminus B$ then there is some $i$ for which $\omega \notin Y_{t_i}^{-1}(A_i)$, and $\omega \in X_{t_i}^{-1}(A_i)$ so $X_{t_i}(\omega )\ne Y_{t_i}(\omega )$. Therefore

$$A\mathrm{△}B\subset \bigcup _{i=1}^n\{\omega \in \mathrm{\Omega }:X_{t_i}(\omega )\ne Y_{t_i}(\omega )\}.$$

Because $X$ is a modification of $Y$, the right-hand side is a union of finitely many $P$-null sets, hence is itself a $P$-null set. $A$ and $B$ each belong to $ℱ$, so $P(A\mathrm{△}B)=0$. Because $P(A\mathrm{△}B)=0$, $P(A)=P(B)$, i.e.

$$P(X_{t_1}\in A_1,\mathrm{\dots },X_{t_n}\in A_n)=P(Y_{t_1}\in A_1,\mathrm{\dots },Y_{t_n}\in A_n).$$

This implies that

$$P_{*}(X_{t_1}\otimes \mathrm{\cdots }\otimes X_{t_n})=P_{*}(Y_{t_1}\otimes \mathrm{\cdots }\otimes Y_{t_n}),$$

namely, $X$ and $Y$ have the same finite-dimensional distributions. ∎

2 Continuous modifications

Let $E$ be a Polish space with Borel $\sigma$-algebra $ℰ$. A stochastic process ${(X_t)}_{t\in {ℝ}_{\ge 0}}$ is called continuous if for each $\omega \in \mathrm{\Omega }$, the path $t\mapsto X_t(\omega )$ is continuous ${ℝ}_{\ge 0}\to E$.

A dyadic rational is an element of

$$D=\bigcup _{i=0}^{\mathrm{\infty }}{2}^{-i}\mathbb{Z}.$$

The Kolmogorov continuity theorem gives conditions under which a stochastic process whose state space is a Polish space has a continuous modification. This is like the Sobolev lemma, which states that if $f\in H^s({ℝ}^d)$ and $s>k+\frac{d}{2}$, then there is some $\varphi \in C^k({ℝ}^d)$ such that $f=\varphi$ almost everywhere. It does not make sense to say that an element of a Sobolev space is itself $C^k$, because elements of Sobolev spaces are equivalence classes of functions, but it does make sense to say that there is a $C^k$ version of this element.

Theorem 2 (Kolmogorov continuity theorem).

Suppose that $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{F}\mathrm{,}P\mathrm{,}{\mathrm{(}{X}_t\mathrm{)}}_{t\mathrm{\in }{\mathrm{R}}_{\mathrm{\ge }\mathrm{0}}}\mathrm{)}$ is a stochastic process with state space ${\mathrm{R}}^d$. If there are $\alpha \mathrm{,}\beta \mathrm{,}c\mathrm{>}\mathrm{0}$ such that

$$E({|X_t-X_s|}^{\alpha })\le c{|t-s|}^{1+\beta },s,t\in {ℝ}_{\ge 0},$$ (1)

then the stochastic process has a continuous modification that itself satisfies (1).

Proof.

Let and let

$$\delta =\beta -\alpha \gamma >0.$$

For $m\ge 1$, let $S_m$ be the set of all pairs $(s,t)$ with

$$s,t\in \{j{2}^{-m}:0\le j\le 2^m\},$$

and $|s-t|=2^{-m}$. There are $2\cdot 2^m$ such pairs, i.e. $|S_m|=2\cdot 2^m$. Let

$$A_m=\bigcup _{(s,t)\in S_m}\{|X_s-X_t|\ge 2^{-\gamma m}\}\in ℱ.$$

For $(s,t)\in S_m$, using Chebyshev’s inequality and (1) we get

$P(|X_t-X_s|\ge 2^{-\gamma m})$ $\le {(2^{\gamma m})}^{\alpha }E({|X_t-X_s|}^{\alpha })$

$\le 2^{\alpha \gamma m}\cdot c{|t-s|}^{1+\beta }$

$=c{2}^{\alpha \gamma m}{2}^{-m(1+\beta )}$

Hence

Because , the Borel-Cantelli lemma tells us that

$$P\left(\bigcap _{n=1}^{\mathrm{\infty }}\bigcup _{m=n}^{\mathrm{\infty }}{A}_m\right)=P(N_0)=0,$$

where for each $\omega \in \mathrm{\Omega }\setminus N_0$ there is some $m_0(\omega )$ such that $\omega \notin A_m$ when $m\ge m_0(\omega )$. That is, for $\omega \in \mathrm{\Omega }\setminus N_0$ there is some $m_0(\omega )$ such that

(2)

Now let $\omega \in \mathrm{\Omega }\setminus N_0$ and let $s,t\in [0,1]$ be dyadic rationals satisfying

Let $m=m(s,t)$ be the greatest integer such that $|s-t|\le 2^{-m}$:

(3)

which implies that $m\ge m_0(\omega )$. There are some $i_0,j_0\in \{0,1,2,3,\mathrm{\dots },2^m\}$ such that

As and , there are sequences ${\sigma }_j,{\tau }_j\in \{0,1\}$, $j>m$, each of which have cofinitely many zero entries, such that

$$s=s_0+\sum _{j>m}{\sigma }_j{2}^{-j},t=t_0+\sum _{j>m}{\tau }_j{2}^{-j}.$$

Because and ,

$$2^{-m}>|(s-s_0)-(t-t_0)|=|(s-t)-(s_0-t_0)|\ge |s_0-t_0|-|s-t|,$$

and with (3),

Thus , so $|i_0-j_0|\in \{0,1\}$ and so either $s_0=t_0$ or $(s_0,t_0)\in S_m$. In the first case, $|X_{t_0}(\omega )-X_{s_0}(\omega )|=0$. In the second case, since $m\ge m_0(\omega )$, by (2) we have

(4)

Define by induction

$$s_n=s_{n-1}+{\sigma }_{m+n}{2}^{-(m+n)},n\ge 1,$$

i.e.

For each $n\ge 1$, $s_n-s_{n-1}\in \{0,2^{-(m+n)}\}$, so either $s_n=s_{n-1}$ or $(s_{n-1},s_n)\in S_{m+n}$, and because $m+n\ge m+1>m_0(\omega )$, applying (2) yields

Because the sequence ${\sigma }_j$ is eventually equal to $0$, the sequence $s_n$ is eventually equal to $s$. Thus

$$\sum _{n=1}^{\mathrm{\infty }}(X_{s_n}(\omega )-X_{s_{n-1}}(\omega ))=X_s(\omega )-X_{s_0}(\omega ),$$

whence

By the same reasoning we get

Using these and (4) yields

$|X_t(\omega )-X_s(\omega )|$ $\le |X_t(\omega )-X_{t_0}(\omega )|+|X_{t_0}(\omega )-X_{s_0}(\omega )|+|X_s(\omega )-X_{s_0}(\omega )|$

$=C\cdot 2^{-\gamma (m+1)},$

for $C=2^{\gamma }+\frac{2}{1-2^{-\gamma }}$. By (3), , hence

$$|X_t(\omega )-X_s(\omega )|\le C{|t-s|}^{\gamma }.$$ (5)

This is true for all dyadic rationals $s,t\in [0,1]$ with $|s-t|\le 2^{-m_0(\omega )}$; when $|s-t|=0$ it is immediate.

For $k\ge 1$, let $X_t^k=X_{k+t}$, which satisfies (1). By what we have worked out, there is a $P$-null set $N_1^{\prime }\in ℱ$ such that for each $\omega \in \mathrm{\Omega }\setminus N_1^{\prime }$ there is some $m_1^{\prime }(\omega )$ such that $m\ge m_1^{\prime }(\omega )$ and $(s,t)\in S_m$ imply that . Let $N_1=N_0\cup N_1^{\prime }$, which is $P$-null, and for $\omega \in \mathrm{\Omega }\setminus N_1$ let $m_1(\omega )=\max \{m_0(\omega ),m_1^{\prime }(\omega )\}$. For $s,t\in D\cap [0,1]$ with $|s-t|\le 2^{-m_1(\omega )}$, what we have worked out yields

$$|X_t(\omega )-X_s(\omega )|\le C{|t-s|}^{\gamma },|X_t^1(\omega )-X_s^1(\omega )|\le C{|t-s|}^{\gamma }.$$

By induction, we get that for each $k\ge 1$ there are $P$-null sets $N_0\subset N_1\subset \mathrm{\cdots }\subset N_k$ and for each $\omega \in \mathrm{\Omega }\setminus N_k$ there is some $m_k(\omega )$ such that for $s,t\in D\cap [0,1]$ with $|s-t|\le 2^{-m_k(\omega )}$,

$|X_t(\omega )-X_s(\omega )|$ $\le C{|t-s|}^{\gamma }$

$|X_t^1(\omega )-X_s^1(\omega )|$ $\le C{|t-s|}^{\gamma }$

$\mathrm{\dots }$

$|X_t^k(\omega )-X_s^k(\omega )|$ $\le C{|t-s|}^{\gamma }.$

Let

$$N_{\gamma }=\bigcup _{k\ge 1}{N}_k,$$

which is an increasing sequence of sets whose union is $P$-null. For $\omega \in \mathrm{\Omega }\setminus N_{\gamma }$, there is a nondecreasing sequence $m_k(\omega )$ such that when $0\le j\le k$ and $s,t\in D\cap [j,j+1]$ with $|s-t|\le 2^{-m_k(\omega )}$, it is the case that $|X_t(\omega )-X_s(\omega )|\le C{|t-s|}^{\gamma }$. For $s,t\in D\cap [0,k+1]$ with $|s-t|\le 2^{-m_k(\omega )}$, because $|s-t|\le \frac{1}{2}$, either there is some $0\le j\le k$ for which $s,t\in [j,j+1]$ or there is some $1\le j\le k$ for which, say, . In the first case, $|X_t(\omega )-X_s(\omega )|\le C{|t-s|}^{\gamma }$. In the second case, because and , we get, because $s,j\in D\cap [j-1,j]$ and $j,t\in D\cap [j,j+1]$,

$|X_t(\omega )-X_s(\omega )|$ $\le |X_t(\omega )-X_j(\omega )|+|X_j(\omega )-X_s(\omega )|$

$\le C{|t-j|}^{\gamma }+C{|j-s|}^{\gamma }$

Thus for

$$C_{\gamma }=2C=2^{\gamma +1}+\frac{4}{1-2^{-\gamma }},$$

we have established that for $\omega \in \mathrm{\Omega }\setminus N_{\gamma }$, $k\ge 1$, and $s,t\in D\cap [0,k+1]$ satisfying $|t-s|\le 2^{-m_k(\omega )}$, it is the case that

$$|X_t(\omega )-X_s(\omega )|\le C_{\gamma }{|t-s|}^{\gamma }.$$ (6)

This implies that for each $\omega \in \mathrm{\Omega }\setminus N_{\gamma }$ and for $k\ge 1$, the mapping $t\mapsto X_t(\omega )$ is uniformly continuous on $D\cap [0,k+1]$. For $t\in {ℝ}_{\ge 0}$ and $\omega \in \mathrm{\Omega }\setminus N_{\gamma }$, define

$$Y_t(\omega )=\underset{\stackrel{s\to t}{s\in D}}{lim}{X}_s(\omega ).$$ (7)

For each $k\ge 0$, because $t\mapsto X_t(\omega )$ is uniformly continuous $D\cap [0,k+1]\to {ℝ}^d$, where $D\cap [0,k+1]$ is dense in $[0,k+1]$ and ${ℝ}^d$ is a complete metric space, the map $t\mapsto Y_t(\omega )$ is uniformly continuous $[0,k+1]\to {ℝ}^d$. Then $t\mapsto Y_t(\omega )$ is continuous ${ℝ}_{\ge 0}\to {ℝ}^d$. For $\omega \in N_{\gamma }$, we define

$$Y_t(\omega )=0,t\in {ℝ}_{\ge 0}.$$

Then for each $\omega \in \mathrm{\Omega }$, $t\mapsto Y_t(\omega )$ is continuous ${ℝ}_{\ge 0}\to {ℝ}^d$. For $t\in {ℝ}_{\ge 0}$, $\omega \mapsto Y_t(\omega )$ is the pointwise limit of the sequence of mappings $\omega \mapsto X_s(\omega )$ as $s\to t$, $s\in D$. For each $s\in D$, $\omega \mapsto X_s(\omega )$ is measurable $ℱ\to {ℬ}_{{ℝ}^d}$, which implies that $\omega \mapsto Y_t(\omega )$ is itself measurable $ℱ\to {ℬ}_{{ℝ}^d}$. Namely, ${(Y_t)}_{t\in {ℝ}_{\ge 0}}$ is a continuous stochastic process.

We must show that $Y$ is a modification of $X$. For $s\in D$, for all $\omega \in \mathrm{\Omega }\setminus N_{\gamma }$ we have $Y_s(\omega )=X_s(\omega )$. For $t\in {ℝ}_{\ge 0}$, there is a sequence $s_n\in D$ tending to $t$, and then for all $\omega \in \mathrm{\Omega }\setminus N_{\gamma }$ by (7) we have $X_{s_n}(\omega )\to Y_t(\omega )$. $P(N_{\gamma })=0$, namely, $X_{s_n}$ converges to $Y_t$ almost surely. Because $X_{s_n}$ converges to $Y_t$ almost surely and $P$ is a probability measure, $X_{s_n}$ converges in measure to $Y_t$. On the other hand, for $\eta >0$, by Chebyshev’s inequality and (1),

$$P\{|X_{s_n}-X_t|\ge \eta \}\le {\eta }^{-\alpha }E(|X_{s_n}-X_t{|}^{\alpha })\le {\eta }^{-\alpha }\cdot c|s_n-t{|}^{1+\beta },$$

and because this is true for each $\eta >0$, this shows that $X_{s_n}$ converges in measure to $X_t$. Hence, the limits $Y_t$ and $X_t$ are equal as equivalence classes of measurable functions $\mathrm{\Omega }\to {ℝ}^d$. That is, $P\{Y_t=X_t\}=1$. This is true for each $t\in {ℝ}_{\ge 0}$, showing that $Y$ is a modification of $X$, completing the proof. ∎

3 Hölder continuity

Let $(X,d)$ and $(Y,\rho )$ be metric spaces, let , and let $\varphi :X\to Y$ be a function. For $x_0\in X$, we say that $\varphi$ is $\gamma$-Hölder continuous at $x_{\mathrm{0}}$ if there is some and some $C_{x_0}$ such that when ,

$$\rho (\varphi (x),\varphi (x_0))\le C_{x_0}d{(x,x_0)}^{\gamma }.$$

We say that $\varphi$ is locally $\gamma$-Hölder continuous if for each $x_0\in X$ there is some and some $C_{x_0}$ such that when and ,

$$\rho (\varphi (x),\varphi (y))\le C_{x_0}d{(x,y)}^{\gamma }.$$

We say that $\varphi$ is uniformly $\gamma$-Hölder continuous if there is some $C$ such that for all $x,y\in X$,

$$\rho (\varphi (x),\varphi (y))\le Cd{(x,y)}^{\gamma }.$$

We establish properties of Hölder continuous functions in the following.

Lemma 3.

Let $V$ be a nonempty subset of ${\mathrm{R}}_{\mathrm{\ge }\mathrm{0}}$, let , and let $f\mathrm{:}V\mathrm{\to }{\mathrm{R}}^d$ be locally $\gamma$-Hölder continuous.

1. If then $f$ is locally ${\gamma }^{\prime }$-Hölder continuous.

2. If $V$ is compact, then $f$ is uniformly $\gamma$-Hölder continuous.

3. If $V$ is an interval of length $T>0$ and there is some $ϵ>0$ and some $C$ such that for all $s,t\in V$ with $|t-s|\le ϵ$ we have

   $$|f(t)-f(s)|\le C{|t-s|}^{\gamma },$$ (8)

   then

   $$|f(t)-f(s)|\le C{\lceil \frac{T}{ϵ}\rceil }^{1-\gamma }{|t-s|}^{\gamma },s,t\in V.$$

Proof.

For $t_0\in {ℝ}_{\ge 0}$, there is some and some $C_{t_0}$ such that when ,

$$|f(t)-f(t_0)|\le C_{t_0}{|t-t_0|}^{\gamma }\le C_{t_0}{|t-t_0|}^{{\gamma }^{\prime }},$$

showing that $f$ is locally ${\gamma }^{\prime }$-Hölder continuous.

With the metric inherited from ${ℝ}_{\ge 0}$, $V$ is a compact metric space. For $t\in V$ and $ϵ>0$, write

which is an open subset of $V$. Because $f$ is locally $\gamma$-Hölder continuous, for each $t\in V$ there is some and some $C_t$ such that for all $u,v\in B_{{ϵ}_t}(t)$,

$$|f(u)-f(v)|\le C_t{|u-v|}^{\gamma }.$$ (9)

Write $U_t=B_{{ϵ}_t}(t)$. Because $t\in U_t$, $\{U_t:t\in V\}$ is an open cover of $V$, and because $V$ is compact there are $t_1,\mathrm{\dots },t_n\in V$ such that $𝔘=\{U_{t_1},\mathrm{\dots },U_{t_n}\}$ is an open cover of $V$. Because $V$ is a compact metric space, there is a Lebesgue number $\delta >0$ of the open cover $𝔘$: for each $t\in V$, there is some $1\le i\le n$ such that $B_{\delta }(t)\subset U_{t_i}$. Let

$$C=\max \{C_{t_1},\mathrm{\dots },C_{t_n},2{\parallel f\parallel }_u{\delta }^{-\gamma }\},$$

For $s,t\in V$ with , i.e. $s\in B_{\delta }(t)$, there is some $1\le i\le n$ with $s,t\in U_{t_i}$. By (9),

$$|f(s)-f(t)|\le C_{t_i}{|s-t|}^{\gamma }\le C{|s-t|}^{\gamma }.$$

On the other hand, for $s,t\in V$ with $|t-s|\ge \delta$,

$$|f(s)-f(t)|\le 2{\parallel f\parallel }_u\le 2{\parallel f\parallel }_u{\left(\frac{|s-t|}{\delta }\right)}^{\gamma }=2{\parallel f\parallel }_u{\delta }^{-\gamma }{|s-t|}^{\gamma }\le C{|s-t|}^{\gamma }.$$

Thus, for all $s,t\in V$,

$$|f(s)-f(t)|\le C{|s-t|}^{\gamma },$$

showing that $f$ is uniformly $\gamma$-Hölder continuous.

Let $n=\lceil \frac{T}{ϵ}\rceil$. For $s,t\in V$, because $V$ is an interval of length $T$, $|s-t|\le T\le ϵn$, and then applying (8), because $\frac{|t-s|}{n}\le ϵ$,

$|f(t)-f(s)|$ $=\left|{\displaystyle \sum _{k=1}^n}f\left(s+(t-s){\displaystyle \frac{k}{n}}\right)-f\left(s+(t-s){\displaystyle \frac{k-1}{n}}\right)\right|$

$\le {\displaystyle \sum _{k=1}^n}\left|f\left(s+(t-s){\displaystyle \frac{k}{n}}\right)-f\left(s+(t-s){\displaystyle \frac{k-1}{n}}\right)\right|$

$\le {\displaystyle \sum _{k=1}^n}C{\left|{\displaystyle \frac{t-s}{n}}\right|}^{\gamma }$

$=C{n}^{1-\gamma }{|t-s|}^{\gamma }.$

∎

The following theorem does not speak about a version of a stochastic process. Rather, it shows what can be said about a stochastic process that satisfies (1) when almost all of its sample paths are continuous.

Theorem 4.

If a stochastic process ${\mathrm{(}{X}_t\mathrm{)}}_{t\mathrm{\in }{\mathrm{R}}_{\mathrm{\ge }\mathrm{0}}}$ with state space ${\mathrm{R}}^d$ satisfies (1) and for almost every $\omega \mathrm{\in }\mathrm{\Omega }$ the map $t\mathrm{\mapsto }{X}_t\mathit{}\mathrm{(}\omega \mathrm{)}$ is continuous ${\mathrm{R}}_{\mathrm{\ge }\mathrm{0}}\mathrm{\to }{\mathrm{R}}^d$, then for almost every $\omega \mathrm{\in }\mathrm{\Omega }$, for every , the map $t\mathrm{\mapsto }{X}_t\mathit{}\mathrm{(}\omega \mathrm{)}$ is locally $\gamma$-Hölder continuous.

Proof.

There is a $P$-null set $N\in ℱ$ such that for $\omega \in \mathrm{\Omega }\setminus N$, the map $t\mapsto X_t(\omega )$ is continuous ${ℝ}_{\ge 0}\to {ℝ}^d$. For each , we have established in (6) that there is a $P$-null set $N_{\gamma }\in ℱ$ such that for $k\ge 1$ there is some $m_k(\omega )$ such that when $s,t\in D\cap [0,k+1]$ and $|t-s|\le 2^{-m_k(\omega )}$,

$$|X_t(\omega )-X_s(\omega )|\le C_{\gamma }{|t-s|}^{\gamma },$$ (10)

where $C_{\gamma }=2^{\gamma +1}+\frac{4}{1-2^{-\gamma }}$. Write $\delta (k,\omega )=2^{-m_k(\omega )}$, and let $M_{\gamma }=N_{\gamma }\cup N$. For $\omega \in \mathrm{\Omega }\setminus M_{\gamma }$, the map $t\mapsto X_t(\omega )$ is continuous ${ℝ}_{\ge 0}\to {ℝ}^d$. For $k\ge 1$ and for $s,t\in [0,k+1]$ satisfying $|s-t|\le \delta (k,\omega )$, say with $s\le t$, let $m=\frac{t-s}{2}$ and let $s\le s_n\le t$ be a sequence of dyadic rationals decreasing to $s$ and let $s\le t_n\le t$ be a sequence of dyadic rationals inceasing to $t$. Then $s_n,t_n\in D\cap [0,k+1]$ and $|s_n-t_n|\le |s-t|\le \delta (k,\omega )$, so by (10),

$$|X_{t_n}(\omega )-X_{s_n}(\omega )|\le C_{\gamma }{|t_n-s_n|}^{\gamma }.$$

Because $\omega \in \mathrm{\Omega }\setminus N$, $X_{t_n}(\omega )\to X_t(\omega )$ and $X_{s_n}(\omega )\to X_s(\omega )$, so

$|X_t(\omega )-X_s(\omega )|$ $\le |X_t(\omega )-X_{t_n}(\omega )|+|X_{t_n}(\omega )-X_{s_n}(\omega )|+|X_s(\omega )-X_{s_n}(\omega )|$

$\le |X_t(\omega )-X_{t_n}(\omega )|+C_{\gamma }{|t_n-s_n|}^{\gamma }+|X_s(\omega )-X_{s_n}(\omega )|$

$\downarrow C_{\gamma }{|t-s|}^{\gamma },$

thus

$$|X_t(\omega )-X_s(\omega )|\le C_{\gamma }{|t-s|}^{\gamma },$$

showing that for and $\omega \in \mathrm{\Omega }\setminus M_{\gamma }$, the map $t\mapsto X_t(\omega )$ is locally $\gamma$-Hölder continuous.

Let be a sequence increasing to $\frac{\beta }{\alpha }$ and let

$$M=\bigcup _{n\ge 1}{M}_{{\gamma }_n},$$

which is a $P$-null set. Let and let $n$ be such that ${\gamma }_n\ge \gamma$. For $\omega \in \mathrm{\Omega }\setminus M$, the map $t\mapsto X_t(\omega )$ is locally ${\gamma }_n$-Hölder continuous, and because $\gamma \le {\gamma }_n$ this implies that the map is locally $\gamma$-Hölder continuous, completing the proof. ∎

Bauer attributes the following theorem to Kolgmorov and Chentsov. It does not merely state that for any there is a modification that is locally $\gamma$-Hölder continuous, but that there is a modification that for all is locally $\gamma$-Hölder continuous.

Theorem 5 (Kolmogorov-Chentsov theorem).

If a stochastic process ${\mathrm{(}{X}_t\mathrm{)}}_{t\mathrm{\in }{\mathrm{R}}_{\mathrm{\ge }\mathrm{0}}}$ with state space ${\mathrm{R}}^d$ satisfies (1), then $X$ has a modification $Y$ such that for all $\omega \mathrm{\in }\mathrm{\Omega }$ and , the path $t\mathrm{\mapsto }{Y}_t\mathit{}\mathrm{(}\omega \mathrm{)}$ is locally $\gamma$-Hölder continuous.

Proof.

Applying the Kolmogorov continuity theorem, there is a continuous modification $Z$ of $X$ that also satisfies (1). By Theorem 4, there is a $P$-null set $M$ such that for $\omega \in \mathrm{\Omega }\setminus M$ and , the map $t\mapsto Z_t(\omega )$ is locally $\gamma$-Hölder continuous. For $t\in {ℝ}_{\ge 0}$, define

i.e. $Y_t=1_{\mathrm{\Omega }\setminus M}{Z}_t$, which is measurable $ℱ\to {ℬ}_{{ℝ}^d}$, and so ${(Y_t)}_{t\in {ℝ}_{\ge 0}}$ is a stochastic process. For every $\omega \in \mathrm{\Omega }$ and , the map $t\mapsto Y_t(\omega )$ is locally $\gamma$-Hölder continuous. For $t\in {ℝ}_{\ge 0}$,

$$\{X_t\ne Y_t\}=\{X_t\ne Y_t,X_t=Z_t\}\cup \{X_t\ne Y_t,X_t\ne Z_t\}\subset \{Y_t\ne Z_t\}\cup \{X_t\ne Z_t\}.$$

Because $P(Y_t\ne Z_t)=P(M)=0$ and $P(X_t\ne Z_t)=0$, since $Z$ is a modification of $X$, we get $P(X_t\ne Y_t)=0$, namely, $Y$ is a modification of $X$. ∎