I am currently working on a basic text on fractional Brownian motion and have come to the point of the (Hölder-)continuity of the sample paths. To me, the Kolmogorov criterion for the continuity of sample paths for stochastic processes is a pretty neat tool to verify the continuity of fractional Brownian motion (that is, its paths). However, what I do not understand so far is if and how this criterion is a direct consequence of the Garsia-Rodemich-Rumsey-Lemma (not the multi-parameter version, I'm talking about the basic one here). Is there anyone who can explain this to me? I have heard my professor say that "having seen the GRR-lemma, there is no necessity for a further proof of Kolmogorov" - do you agree on this question and if not, what remains to be done?
Very thankful for any advice!
Your professor is gliding past a subtle point.
Let $X=(X_t)_{0\le t\le 1}$ be a stochastic process defined on the probability space $(\Omega,\mathcal F,P)$. The Kolmogorov criterion says that if there are constants $\alpha>0$, $\beta>0$, $0<C<\infty$ such that $$ E|X_t-X_s|^\alpha\le C|t-s|^{1+\beta},\qquad\forall s,t\in[0,1], $$ then there is a modification $(\tilde X_t)_{0\le t\le 1}$ of $X$ (that is, $P[\tilde X_t\not=X_t]=0$ for all $t\in[0,1]$) such that, with probability $1$, $t\mapsto \tilde X_t$ is Hölder continuous of order $\gamma$ for each $\gamma\in(0,\beta/\alpha)$.
The GRR lemma is a real-variable result, to be applied to a sample path $t\mapsto X_t(\omega)$ for $P$-a.e. $\omega\in\Omega$. The lemma (stated in a somewhat simplified form) is as follows: Let $\Psi$ and $p$ be strictly increasing continuous functions on the half line, with $\Psi(0)=p(0)=0$ and $\lim_{x\to+\infty}\Psi(x)=+\infty$. Let $f:[0,1]\to\Bbb R$ be a continuous function such that $$ B:=\int_0^1\int_0^1\Psi\left(|f(s)-f(t)|/p(|s-t|)\right)\,ds\,dt<\infty. $$ Then $$ |f(s)-f(t)|\le8\int_0^{|s-t|}\Psi^{-1}(4B/u^2)\,dp(u),\qquad\forall s,t\in[0,1]. $$ One can think to deduce the Kolmogorov criterion from the GRR lemma be taking $\Psi(u):=u^\alpha$ and $p(u) = u^b$, where $b\in(0,(2+\beta)/\alpha)$. It's not hard to check using Fubini that if the Kolmogorov criterion is satisfied then $$ E\left(\int_0^1\int_0^1\Psi\left(|X_s-X_t|/p(|s-t|)\right)\,ds\,dt\right)<\infty, $$ so that $$ B(\omega):=\int_0^1\int_0^1\Psi\left(|X_s(\omega)-X_t(\omega)|/p(|s-t|)\right)\,ds\,dt<\infty, \hbox{ a.s.} $$ permitting the application of the GRR lemma with the indicated choices of $\Psi$ and $p$, to $P$-a.e. sample path.
The subtle point here is that to apply the GRR lemma one needs to already know that $t\mapsto X_t(\omega)$ is a continuous function, which is after all (part of) what we are trying to prove is a consequence of the Kolmogorov moment condition. One way out of this bit of circularity is as follows. For $n=1,2,\ldots$, restrict the time parameter of $(X_t)$ to the rank-$n$ dyadic rationals $D_n:=\{k2^{-n}:k=0,1,\ldots,2^n\}$ and then linearly interpolate the values of $X_t$ at intermediate times to obtain a continuous process $(X^{(n)}_t)_{0\le t\le 1}$ that agrees with $X$ at times in $D_n$. The interpolated processes satisfy the Kolmogorov moment condition with a slightly worse constant $C$ ($3^\alpha C$ will do). The GRR lemma applies to each $X^{(n)}$, and with a little care this leads to the conclusion that there is a random variable $B(\omega)\ge 0$ such that $P(B<\infty)=1$ and $$ |X_q(\omega)-X_r(\omega)|\le B(\omega)|q-r|^\gamma,\qquad \forall q,r\in D:=\cup_n D_n, $$ for $P$-a.e. $\omega$. (Here $\gamma\in(0,\beta/\alpha)$ as before.) Thus, with probability $1$, $t\mapsto X_t(\omega)$ restricted to the dyadic rationals is Hölder continuous, and so admits a Hölder continuous extension, call it $\tilde X_t(\omega)$, to all of $[0,1]$. As the Kolmogorov moment condition obviously implies that $X$ is stochastically continuous, we have $P[\tilde X_t\not= X_t]=0$ for each $t$. That is, $\tilde X$ is a Hölder continuous modification of $X$. For the details of such an argument see section 2.1 (pages 46--51) in the book Multidimensional Diffusion processes by Stroock and Varadhan.
It may be that a form of the GRR lemma is true with the hypothesis of continuity on $f$ relaxed to measurability, the conclusion being weakend to the assertion that $f$ is a.e. equal to a continuous function $\tilde f$ satisfying the conclusion of the original GRR lemma. Note that if a process $X$ satisfies the Kolmogorov moment condition, then $X$ is stochastically continuous, so by a result of Doob $X$ admits a (separable and) measurable modification.