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

Jordan Bell
June 11, 2015

1 Modifications

Let $(\Omega,\mathscr{F},P)$ be a probability space, let $I$ be a nonempty set, and let $(E,\mathscr{E})$ 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 $X_{t}:(\Omega,\mathscr{F})\to(E,\mathscr{E})$. If $X$ and $Y$ are stochastic processes, we say that $X$ is a modification of $Y$ if for each $t\in I$,

 $P\{\omega\in\Omega:X_{t}(\omega)=Y_{t}(\omega)\}=1.$
Lemma 1.

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

Proof.

For $t_{1},\ldots,t_{n}\in I$, let $A_{i}\in\mathscr{E}$ for each $1\leq i\leq n$, and let

 $A=\bigcap_{i=1}^{n}X_{t_{i}}^{-1}(A_{i})\in\mathscr{F},\qquad B=\bigcap_{i=1}^% {n}Y_{t_{i}}^{-1}(A_{i})\in\mathscr{F}.$

If $\omega\in A\setminus B$ then there is some $i$ for which $\omega\not\in Y_{t_{i}}^{-1}(A_{i})$, and $\omega\in X_{t_{i}}^{-1}(A_{i})$ so $X_{t_{i}}(\omega)\neq Y_{t_{i}}(\omega)$. Therefore

 $A\triangle B\subset\bigcup_{i=1}^{n}\{\omega\in\Omega:X_{t_{i}}(\omega)\neq Y_% {t_{i}}(\omega)\}.$

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 $\mathscr{F}$, so $P(A\triangle B)=0$.11 1 We have not assumed that $(\Omega,\mathscr{F},P)$ is a complete measure space, so we must verify that a set is measurable before speaking about its measure. Because $P(A\triangle B)=0$, $P(A)=P(B)$, i.e.

 $P(X_{t_{1}}\in A_{1},\ldots,X_{t_{n}}\in A_{n})=P(Y_{t_{1}}\in A_{1},\ldots,Y_% {t_{n}}\in A_{n}).$

This implies that2

 $P_{*}(X_{t_{1}}\otimes\cdots\otimes X_{t_{n}})=P_{*}(Y_{t_{1}}\otimes\cdots% \otimes Y_{t_{n}}),$

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

2 Continuous modifications

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

A dyadic rational is an element of

 $D=\bigcup_{i=0}^{\infty}2^{-i}\mathbb{Z}.$

The Kolmogorov continuity theorem gives conditions under which a stochastic process whose state space is a Polish space has a continuous modification.33 3 Heinz Bauer, Probability Theory, p. 335, Theorem 39.3. It was only after working through the proof given by Bauer that I realized that the statement is true when the state space is a Polish space rather than merely $\mathbb{R}^{d}$. In the proof I do not use that $|\cdot|$ is a norm on $\mathbb{R}^{d}$, and only use that $d(x,y)=|x-y|$ is a metric on $\mathbb{R}^{d}$, so it is straightforward to rewrite the proof. This is like the Sobolev lemma,44 4 Walter Rudin, Functional Analysis, second ed., p. 202, Theorem 7.25. which states that if $f\in H^{s}(\mathbb{R}^{d})$ and $s>k+\frac{d}{2}$, then there is some $\phi\in C^{k}(\mathbb{R}^{d})$ such that $f=\phi$ 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 $(\Omega,\mathscr{F},P,(X_{t})_{t\in\mathbb{R}_{\geq 0}})$ is a stochastic process with state space $\mathbb{R}^{d}$. If there are $\alpha,\beta,c>0$ such that

 $E(|X_{t}-X_{s}|^{\alpha})\leq c|t-s|^{1+\beta},\qquad s,t\in\mathbb{R}_{\geq 0},$ (1)

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

Proof.

Let $0<\gamma<\frac{\beta}{\alpha}$ and let

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

For $m\geq 1$, let $S_{m}$ be the set of all pairs $(s,t)$ with

 $s,t\in\{j2^{-m}:0\leq j\leq 2^{m}\},$

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

 $A_{m}=\bigcup_{(s,t)\in S_{m}}\{|X_{s}-X_{t}|\geq 2^{-\gamma m}\}\in\mathscr{F}.$

For $(s,t)\in S_{m}$, using Chebyshev’s inequality and (1) we get

 $\displaystyle P(|X_{t}-X_{s}|\geq 2^{-\gamma m})$ $\displaystyle\leq(2^{\gamma m})^{\alpha}E(|X_{t}-X_{s}|^{\alpha})$ $\displaystyle\leq 2^{\alpha\gamma m}\cdot c|t-s|^{1+\beta}$ $\displaystyle=c2^{\alpha\gamma m}2^{-m(1+\beta)}$ $\displaystyle

Hence

 $P(A_{m})\leq\sum_{(s,t)\in S_{m}}P\{|X_{s}-X_{t}|\geq 2^{-\gamma m}\}<\sum_{(s% ,t)\in S_{m}}c2^{-m-\delta m}=2c\cdot 2^{-\delta m}.$

Because $\sum_{m}P(A_{m})<\infty$, the Borel-Cantelli lemma tells us that

 $P\left(\bigcap_{n=1}^{\infty}\bigcup_{m=n}^{\infty}A_{m}\right)=P(N_{0})=0,$

where for each $\omega\in\Omega\setminus N_{0}$ there is some $m_{0}(\omega)$ such that $\omega\not\in A_{m}$ when $m\geq m_{0}(\omega)$. That is, for $\omega\in\Omega\setminus N_{0}$ there is some $m_{0}(\omega)$ such that

 $|X_{t}(\omega)-X_{s}(\omega)|<2^{-\gamma m},\qquad m\geq m_{0}(\omega),\quad(s% ,t)\in S_{m}.$ (2)

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

 $0<|s-t|\leq 2^{-m_{0}(\omega)}.$

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

 $2^{-m-1}<|s-t|\leq 2^{-m},$ (3)

which implies that $m\geq m_{0}(\omega)$. There are some $i_{0},j_{0}\in\{0,1,2,3,\ldots,2^{m}\}$ such that

 $s_{0}=i_{0}2^{-m}\leq s<(i_{0}+1)2^{-m},\qquad t_{0}=j_{0}2^{-m}\leq t<(j_{0}+% 1)2^{-m}.$

As $0\leq s-s_{0}<2^{-m}$ and $0\leq t-t_{0}<2^{-m}$, there are sequences $\sigma_{j},\tau_{j}\in\{0,1\}$, $j>m$, each of which have cofinitely many zero entries, such that

 $s=s_{0}+\sum_{j>m}\sigma_{j}2^{-j},\qquad t=t_{0}+\sum_{j>m}\tau_{j}2^{-j}.$

Because $0\leq s-s_{0}<2^{-m}$ and $\leq t-t_{0}<2^{-m}$,

 $2^{-m}>|(s-s_{0})-(t-t_{0})|=|(s-t)-(s_{0}-t_{0})|\geq|s_{0}-t_{0}|-|s-t|,$

and with (3),

 $|s_{0}-t_{0}|<2^{-m}+|s-t|\leq 2^{-m}+2^{-m}=2^{-m+1}.$

Thus $|i_{0}-j_{0}|<2$, 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 $m\geq m_{0}(\omega)$, by (2) we have

 $|X_{t_{0}}(\omega)-X_{s_{0}}(\omega)|<2^{-\gamma m}.$ (4)

Define by induction

 $s_{n}=s_{n-1}+\sigma_{m+n}2^{-(m+n)},\qquad n\geq 1,$

i.e.

 $s_{n}=s_{0}+\sum_{m

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

 $|X_{s_{n}}(\omega)-X_{s_{n-1}}(\omega)|<2^{-\gamma(m+n)}.$

Because the sequence $\sigma_{j}$ is eventually equal to $0$, the sequence $s_{n}$ is eventually equal to $s$. Thus

 $\sum_{n=1}^{\infty}(X_{s_{n}}(\omega)-X_{s_{n-1}}(\omega))=X_{s}(\omega)-X_{s_% {0}}(\omega),$

whence

 $|X_{s}(\omega)-X_{s_{0}}(\omega)|\leq\sum_{n=1}^{\infty}|X_{s_{n}}(\omega)-X_{% s_{n-1}}(\omega)|<\sum_{n=1}^{\infty}2^{-\gamma(m+n)}=\frac{2^{-\gamma(m+1)}}{% 1-2^{-\gamma}}.$

By the same reasoning we get

 $|X_{t}(\omega)-X_{t_{0}}(\omega)|<\frac{2^{-\gamma(m+1)}}{1-2^{-\gamma}}.$

Using these and (4) yields

 $\displaystyle|X_{t}(\omega)-X_{s}(\omega)|$ $\displaystyle\leq|X_{t}(\omega)-X_{t_{0}}(\omega)|+|X_{t_{0}}(\omega)-X_{s_{0}% }(\omega)|+|X_{s}(\omega)-X_{s_{0}}(\omega)|$ $\displaystyle<\frac{2^{-\gamma(m+1)}}{1-2^{-\gamma}}+2^{-\gamma m}+\frac{2^{-% \gamma(m+1)}}{1-2^{-\gamma}}$ $\displaystyle=C\cdot 2^{-\gamma(m+1)},$

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

 $|X_{t}(\omega)-X_{s}(\omega)|\leq C|t-s|^{\gamma}.$ (5)

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

For $k\geq 1$, let $X^{k}_{t}=X_{k+t}$, which satisfies (1). By what we have worked out, there is a $P$-null set $N_{1}^{\prime}\in\mathscr{F}$ such that for each $\omega\in\Omega\setminus N_{1}^{\prime}$ there is some $m_{1}^{\prime}(\omega)$ such that $m\geq m_{1}^{\prime}(\omega)$ and $(s,t)\in S_{m}$ imply that $|X^{1}_{t}(\omega)-X^{1}_{s}(\omega)|<2^{-\gamma m}$. Let $N_{1}=N_{0}\cup N_{1}^{\prime}$, which is $P$-null, and for $\omega\in\Omega\setminus N_{1}$ let $m_{1}(\omega)=\max\{m_{0}(\omega),m_{1}^{\prime}(\omega)\}$. For $s,t\in D\cap[0,1]$ with $|s-t|\leq 2^{-m_{1}(\omega)}$, what we have worked out yields

 $|X_{t}(\omega)-X_{s}(\omega)|\leq C|t-s|^{\gamma},\qquad|X^{1}_{t}(\omega)-X^{% 1}_{s}(\omega)|\leq C|t-s|^{\gamma}.$

By induction, we get that for each $k\geq 1$ there are $P$-null sets $N_{0}\subset N_{1}\subset\cdots\subset N_{k}$ and for each $\omega\in\Omega\setminus N_{k}$ there is some $m_{k}(\omega)$ such that for $s,t\in D\cap[0,1]$ with $|s-t|\leq 2^{-m_{k}(\omega)}$,

 $\displaystyle|X_{t}(\omega)-X_{s}(\omega)|$ $\displaystyle\leq C|t-s|^{\gamma}$ $\displaystyle|X^{1}_{t}(\omega)-X^{1}_{s}(\omega)|$ $\displaystyle\leq C|t-s|^{\gamma}$ $\displaystyle\ldots$ $\displaystyle|X^{k}_{t}(\omega)-X^{k}_{s}(\omega)|$ $\displaystyle\leq C|t-s|^{\gamma}.$

Let

 $N_{\gamma}=\bigcup_{k\geq 1}N_{k},$

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

 $\displaystyle|X_{t}(\omega)-X_{s}(\omega)|$ $\displaystyle\leq|X_{t}(\omega)-X_{j}(\omega)|+|X_{j}(\omega)-X_{s}(\omega)|$ $\displaystyle\leq C|t-j|^{\gamma}+C|j-s|^{\gamma}$ $\displaystyle<2C|t-s|^{\gamma}.$

Thus for

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

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

 $|X_{t}(\omega)-X_{s}(\omega)|\leq C_{\gamma}|t-s|^{\gamma}.$ (6)

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

 $Y_{t}(\omega)=\lim_{\stackrel{{\scriptstyle s\to t}}{{s\in D}}}X_{s}(\omega).$ (7)

For each $k\geq 0$, because $t\mapsto X_{t}(\omega)$ is uniformly continuous $D\cap[0,k+1]\to\mathbb{R}^{d}$, where $D\cap[0,k+1]$ is dense in $[0,k+1]$ and $\mathbb{R}^{d}$ is a complete metric space, the map $t\mapsto Y_{t}(\omega)$ is uniformly continuous $[0,k+1]\to\mathbb{R}^{d}$.55 5 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 77, Lemma 3.11. Then $t\mapsto Y_{t}(\omega)$ is continuous $\mathbb{R}_{\geq 0}\to\mathbb{R}^{d}$. For $\omega\in N_{\gamma}$, we define

 $Y_{t}(\omega)=0,\qquad t\in\mathbb{R}_{\geq 0}.$

Then for each $\omega\in\Omega$, $t\mapsto Y_{t}(\omega)$ is continuous $\mathbb{R}_{\geq 0}\to\mathbb{R}^{d}$. For $t\in\mathbb{R}_{\geq 0}$, $\omega\mapsto Y_{t}(\omega)$ 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 $\mathscr{F}\to\mathscr{B}_{\mathbb{R}^{d}}$, which implies that $\omega\mapsto Y_{t}(\omega)$ is itself measurable $\mathscr{F}\to\mathscr{B}_{\mathbb{R}^{d}}$.66 6 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 142, Lemma 4.29. Namely, $(Y_{t})_{t\in\mathbb{R}_{\geq 0}}$ is a continuous stochastic process.

We must show that $Y$ is a modification of $X$. For $s\in D$, for all $\omega\in\Omega\setminus N_{\gamma}$ we have $Y_{s}(\omega)=X_{s}(\omega)$. For $t\in\mathbb{R}_{\geq 0}$, there is a sequence $s_{n}\in D$ tending to $t$, and then for all $\omega\in\Omega\setminus N_{\gamma}$ 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 $Y_{t}$.77 7 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 479, Theorem 13.37. On the other hand, for $\eta>0$, by Chebyshev’s inequality and (1),

 $P\{|X_{s_{n}}-X_{t}|\geq\eta\}\leq\eta^{-\alpha}E(|X_{s_{n}}-X_{t}|^{\alpha})% \leq\eta^{-\alpha}\cdot c|s_{n}-t|^{1+\beta},$

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 $\Omega\to\mathbb{R}^{d}$.8 That is, $P\{Y_{t}=X_{t}\}=1$. This is true for each $t\in\mathbb{R}_{\geq 0}$, showing that $Y$ is a modification of $X$, completing the proof. ∎

3 Hölder continuity

Let $(X,d)$ and $(Y,\rho)$ be metric spaces, let $0<\gamma<1$, and let $\phi:X\to Y$ be a function. For $x_{0}\in X$, we say that $\phi$ is $\gamma$-Hölder continuous at $x_{0}$ if there is some $0<\epsilon_{x_{0}}<1$ and some $C_{x_{0}}$ such that when $d(x,x_{0})<\epsilon_{x_{0}}$,

 $\rho(\phi(x),\phi(x_{0}))\leq C_{x_{0}}d(x,x_{0})^{\gamma}.$

We say that $\phi$ is locally $\gamma$-Hölder continuous if for each $x_{0}\in X$ there is some $0<\epsilon_{x_{0}}<1$ and some $C_{x_{0}}$ such that when $d(x,x_{0})<\epsilon_{x_{0}}$ and $d(y,x_{0})<\epsilon_{x_{0}}$,

 $\rho(\phi(x),\phi(y))\leq C_{x_{0}}d(x,y)^{\gamma}.$

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

 $\rho(\phi(x),\phi(y))\leq Cd(x,y)^{\gamma}.$

We establish properties of Hölder continuous functions in the following.99 9 Achim Klenke, Probability Theory: A Comprehensive Course, p. 448, Lemma 21.3.

Lemma 3.

Let $V$ be a nonempty subset of $\mathbb{R}_{\geq 0}$, let $0<\gamma<1$, and let $f:V\to\mathbb{R}^{d}$ be locally $\gamma$-Hölder continuous.

1. 1.

If $0<\gamma^{\prime}<\gamma$ then $f$ is locally $\gamma^{\prime}$-Hölder continuous.

2. 2.

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

3. 3.

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

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

then

 $|f(t)-f(s)|\leq C\left\lceil\frac{T}{\epsilon}\right\rceil^{1-\gamma}|t-s|^{% \gamma},\qquad s,t\in V.$
Proof.

For $t_{0}\in\mathbb{R}_{\geq 0}$, there is some $0<\epsilon_{t_{0}}<1$ and some $C_{t_{0}}$ such that when $|t-t_{0}|<\epsilon_{t_{0}}$,

 $|f(t)-f(t_{0})|\leq C_{t_{0}}|t-t_{0}|^{\gamma}\leq C_{t_{0}}|t-t_{0}|^{\gamma% ^{\prime}},$

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

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

 $B_{\epsilon}(t)=\{v\in V:|v-t|<\epsilon\},$

which is an open subset of $V$. Because $f$ is locally $\gamma$-Hölder continuous, for each $t\in V$ there is some $0<\epsilon_{t}<1$ and some $C_{t}$ such that for all $u,v\in B_{\epsilon_{t}}(t)$,

 $|f(u)-f(v)|\leq C_{t}|u-v|^{\gamma}.$ (9)

Write $U_{t}=B_{\epsilon_{t}}(t)$. Because $t\in U_{t}$, $\{U_{t}:t\in V\}$ is an open cover of $V$, and because $V$ is compact there are $t_{1},\ldots,t_{n}\in V$ such that $\mathfrak{U}=\{U_{t_{1}},\ldots,U_{t_{n}}\}$ is an open cover of $V$. Because $V$ is a compact metric space, there is a Lebesgue number $\delta>0$ of the open cover $\mathfrak{U}$:1010 10 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 85, Lemma 3.27. for each $t\in V$, there is some $1\leq i\leq n$ such that $B_{\delta}(t)\subset U_{t_{i}}$. Let

 $C=\max\{C_{t_{1}},\ldots,C_{t_{n}},2\left\|f\right\|_{u}\delta^{-\gamma}\},$

For $s,t\in V$ with $|t-s|<\delta$, i.e. $s\in B_{\delta}(t)$, there is some $1\leq i\leq n$ with $s,t\in U_{t_{i}}$. By (9),

 $|f(s)-f(t)|\leq C_{t_{i}}|s-t|^{\gamma}\leq C|s-t|^{\gamma}.$

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

 $|f(s)-f(t)|\leq 2\left\|f\right\|_{u}\leq 2\left\|f\right\|_{u}\left(\frac{|s-% t|}{\delta}\right)^{\gamma}=2\left\|f\right\|_{u}\delta^{-\gamma}|s-t|^{\gamma% }\leq C|s-t|^{\gamma}.$

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

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

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

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

 $\displaystyle|f(t)-f(s)|$ $\displaystyle=\left|\sum_{k=1}^{n}f\left(s+(t-s)\frac{k}{n}\right)-f\left(s+(t% -s)\frac{k-1}{n}\right)\right|$ $\displaystyle\leq\sum_{k=1}^{n}\left|f\left(s+(t-s)\frac{k}{n}\right)-f\left(s% +(t-s)\frac{k-1}{n}\right)\right|$ $\displaystyle\leq\sum_{k=1}^{n}C\left|\frac{t-s}{n}\right|^{\gamma}$ $\displaystyle=Cn^{1-\gamma}|t-s|^{\gamma}.$

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.1111 11 Heinz Bauer, Probability Theory, p. 338, Theorem 39.4.

Theorem 4.

If a stochastic process $(X_{t})_{t\in\mathbb{R}_{\geq 0}}$ with state space $\mathbb{R}^{d}$ satisfies (1) and for almost every $\omega\in\Omega$ the map $t\mapsto X_{t}(\omega)$ is continuous $\mathbb{R}_{\geq 0}\to\mathbb{R}^{d}$, then for almost every $\omega\in\Omega$, for every $0<\gamma<\frac{\beta}{\alpha}$, the map $t\mapsto X_{t}(\omega)$ is locally $\gamma$-Hölder continuous.

Proof.

There is a $P$-null set $N\in\mathscr{F}$ such that for $\omega\in\Omega\setminus N$, the map $t\mapsto X_{t}(\omega)$ is continuous $\mathbb{R}_{\geq 0}\to\mathbb{R}^{d}$. For each $0<\gamma<\frac{\beta}{\alpha}$, we have established in (6) that there is a $P$-null set $N_{\gamma}\in\mathscr{F}$ such that for $k\geq 1$ there is some $m_{k}(\omega)$ such that when $s,t\in D\cap[0,k+1]$ and $|t-s|\leq 2^{-m_{k}(\omega)}$,

 $|X_{t}(\omega)-X_{s}(\omega)|\leq C_{\gamma}|t-s|^{\gamma},$ (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 $\omega\in\Omega\setminus M_{\gamma}$, the map $t\mapsto X_{t}(\omega)$ is continuous $\mathbb{R}_{\geq 0}\to\mathbb{R}^{d}$. For $k\geq 1$ and for $s,t\in[0,k+1]$ satisfying $|s-t|\leq\delta(k,\omega)$, say with $s\leq t$, let $m=\frac{t-s}{2}$ and let $s\leq s_{n}\leq t$ be a sequence of dyadic rationals decreasing to $s$ and let $s\leq t_{n}\leq t$ be a sequence of dyadic rationals inceasing to $t$. Then $s_{n},t_{n}\in D\cap[0,k+1]$ and $|s_{n}-t_{n}|\leq|s-t|\leq\delta(k,\omega)$, so by (10),

 $|X_{t_{n}}(\omega)-X_{s_{n}}(\omega)|\leq C_{\gamma}|t_{n}-s_{n}|^{\gamma}.$

Because $\omega\in\Omega\setminus N$, $X_{t_{n}}(\omega)\to X_{t}(\omega)$ and $X_{s_{n}}(\omega)\to X_{s}(\omega)$, so

 $\displaystyle|X_{t}(\omega)-X_{s}(\omega)|$ $\displaystyle\leq|X_{t}(\omega)-X_{t_{n}}(\omega)|+|X_{t_{n}}(\omega)-X_{s_{n}% }(\omega)|+|X_{s}(\omega)-X_{s_{n}}(\omega)|$ $\displaystyle\leq|X_{t}(\omega)-X_{t_{n}}(\omega)|+C_{\gamma}|t_{n}-s_{n}|^{% \gamma}+|X_{s}(\omega)-X_{s_{n}}(\omega)|$ $\displaystyle\downarrow C_{\gamma}|t-s|^{\gamma},$

thus

 $|X_{t}(\omega)-X_{s}(\omega)|\leq C_{\gamma}|t-s|^{\gamma},$

showing that for $0<\gamma<\frac{\beta}{\alpha}$ and $\omega\in\Omega\setminus M_{\gamma}$, the map $t\mapsto X_{t}(\omega)$ is locally $\gamma$-Hölder continuous.

Let $0<\gamma_{n}<\frac{\beta}{\alpha}$ be a sequence increasing to $\frac{\beta}{\alpha}$ and let

 $M=\bigcup_{n\geq 1}M_{\gamma_{n}},$

which is a $P$-null set. Let $0<\gamma<\frac{\beta}{\alpha}$ and let $n$ be such that $\gamma_{n}\geq\gamma$. For $\omega\in\Omega\setminus M$, the map $t\mapsto X_{t}(\omega)$ is locally $\gamma_{n}$-Hölder continuous, and because $\gamma\leq\gamma_{n}$ this implies that the map is locally $\gamma$-Hölder continuous, completing the proof. ∎

Bauer attributes the following theorem to Kolgmorov and Chentsov.1212 12 Nikolai Nikolaevich Chentsov, 1930–1993, obituary in Russian Math. Surveys 48 (1993), no. 2, 161–166. It does not merely state that for any $0<\gamma<\frac{\beta}{\alpha}$ there is a modification that is locally $\gamma$-Hölder continuous, but that there is a modification that for all $0<\gamma<\frac{\beta}{\alpha}$ is locally $\gamma$-Hölder continuous.1313 13 Heinz Bauer, Probability Theory, p. 339, Corollary 39.5.

Theorem 5 (Kolmogorov-Chentsov theorem).

If a stochastic process $(X_{t})_{t\in\mathbb{R}_{\geq 0}}$ with state space $\mathbb{R}^{d}$ satisfies (1), then $X$ has a modification $Y$ such that for all $\omega\in\Omega$ and $0<\gamma<\frac{\beta}{\alpha}$, the path $t\mapsto Y_{t}(\omega)$ 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 $\omega\in\Omega\setminus M$ and $0<\gamma<\frac{\beta}{\alpha}$, the map $t\mapsto Z_{t}(\omega)$ is locally $\gamma$-Hölder continuous. For $t\in\mathbb{R}_{\geq 0}$, define

 $Y_{t}(\omega)=\begin{cases}Z_{t}(\omega)&\omega\in\Omega\setminus M\\ 0&\omega\in M,\end{cases}$

i.e. $Y_{t}=1_{\Omega\setminus M}Z_{t}$, which is measurable $\mathscr{F}\to\mathscr{B}_{\mathbb{R}^{d}}$, and so $(Y_{t})_{t\in\mathbb{R}_{\geq 0}}$ is a stochastic process. For every $\omega\in\Omega$ and $0<\gamma<\frac{\beta}{\alpha}$, the map $t\mapsto Y_{t}(\omega)$ is locally $\gamma$-Hölder continuous. For $t\in\mathbb{R}_{\geq 0}$,

 $\{X_{t}\neq Y_{t}\}=\{X_{t}\neq Y_{t},X_{t}=Z_{t}\}\cup\{X_{t}\neq Y_{t},X_{t}% \neq Z_{t}\}\subset\{Y_{t}\neq Z_{t}\}\cup\{X_{t}\neq Z_{t}\}.$

Because $P(Y_{t}\neq Z_{t})=P(M)=0$ and $P(X_{t}\neq Z_{t})=0$, since $Z$ is a modification of $X$, we get $P(X_{t}\neq Y_{t})=0$, namely, $Y$ is a modification of $X$. ∎