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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 $\gamma\in\big[0,{\alpha H\over \alpha}\big)$?

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In fact, there Kolmogorov criterion in your formulation can not help apriori. Use this one galton.uchicago.edu/~lalley/Courses/385/GaussianProcesses.pdf on the page six –  Tarasenya Dec 17 '12 at 22:17
or for instance, here statslab.cam.ac.uk/~beresty/teach/StoCal/sc3.pdf page 10. Good lecture notes, by the way. –  Tarasenya Dec 17 '12 at 22:22

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