I would like to prove the following theorem:

Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$.

For any $\alpha >0$ we have by self similarity $$\mathbb{E}[|B_H(t)-B_H(s)|^{\alpha}]=\mathbb{E}[|B_H(1)|^{\alpha}]|t-s|^{\alpha H}.$$

The proof is done after applying the criterion of Kolmogorov which says:

A process $(X_t)_{t\in\mathbb{R}}$ admits a continuous modification if there exist constants $a,b,k>0$ such that $$\mathbb{E}[|X(t)-X(s)|^a]\leq k|t-s|^{1+b}$$ for all $s,t\in\mathbb{R}$.

But I don't know how to apply this criterion. Any help please.


Well, maybe I should try to state my problem more precisely. I just would like to understand the proof. If I use the criterion of Kolmogorov it should hold $$\mathbb{E}[|B_H(t)-B_H(s)|^{\alpha}]=\mathbb{E}[|B_H(1)|^{\alpha}]|t-s|^{\alpha H}\leq k|t-s|^{1+\beta}$$ for $\alpha,\beta,k>0$, right? I don't see any relationship to the Hölder continuity.

Is there nobody who can demonstrate the proof for me to understand?

Maybe I should set $k=\mathbb{E}[|B_H(1)|^{\alpha}]$ and $\beta=\alpha H$ and say that $B_H$ is $\gamma$-Hölder continuous for every $\gamma\in\big[0,{\alpha H\over \alpha}\big)$?


There exists an extension of Kolmogorov's continuity criterion.

Theorem 1.4. of these lecture notes by Nathanaël Berestycki states the following:

Theorem. Let $(X_t)_{t\in[0,1]}$ be a real-valued stochastic process. Suppose that there exists positive constants $p,c,\varepsilon$ such that for each $s,t\in[0,1]$, $$\mathbb E\left[\left|X_t-X_s\right|^p\right]\leqslant c\left|t-s\right|^{1+\varepsilon}.$$ Then there exists a modification $\widetilde X$ of $X$ which is almost surely $\alpha$-Hölder continuous for any $\alpha\in (0,\varepsilon /p)$.

When $X=B_H$, we have
$$\mathbb E\left[\left|X_t-X_s\right|^p\right]=\left|t-s\right|^{pH}\mathbb E\left[\left|B_H(1)\right|^p\right],$$ hence we can use the theorem with $p:=(1+R)/H$ (for a fixed $R$) $c:=\mathbb E\left[\left|B_H(1)\right|^p\right]$, $\varepsilon:=pH-1=R $, which provides an $\alpha$-Hölder continuous modification for each $$\alpha\lt \frac{\varepsilon}p= H\frac{R}{1+R} .$$ As $R$ is arbitrary, we can get a modification for any exponent strictly smaller than $H$.


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