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Let $0<H<1.$ a real-valued Gaussian process $(B_H(t))_{t\ge 0}$ is called fractional Brownian motion (fBm) if $E(B_H(t))=0$ and
$$E(B_H(t)B_H(s))=\frac{1}{2}(t^{2H}+s^{2H}-|t-s|^{2H}).\quad (*)$$
Fact 1: $B_H$ is self-similar, i.e., $B_H(at)\overset{d}{=}|a|^HB_H(t).$
Proof. For any $a>0$, by (*) we have
$$E(B_H(at)B_H(as))=\frac{1}{2}[(at)^{2H}+(as)^{2H}-(a|t-s|)^{2H}]=E[(a^HB_H(t))(a^HB_H(s))].$$
Since $E(B_H(at))=E(a^HB_H(t))=0$ and the above equality implies that $(B_H(at))$ and $(a^HB_H(t))$ have the same covariance matrix. Hence, $B_H(at)\overset{d}{=}|a|^HB_H(t).$
Fact 2: $B_H(t)$ has staionary increnments, i.e., $B_H(t)-B_H(s)\overset{d}{=}B_H(t-s).$
Proof: the proof is similar to that of Fact 1.
Fact 3: $B_H$ admits a version whose sample paths are amlost surely holder continuous of order strict less than $H$, i.e., for each sample path, for any $\varepsilon>0$, there exists a constant $c$ such that
$$|B_H(t)-B_H(s)|\le c |t-s|^{H-\varepsilon}, s,t>0.$$
Proof. We need the following Kolmogorov’s continuity criterion.
Theorem (Kolmogorov’s continuity criterion): Let $\alpha, \varepsilon, c>0$. If a $d$-dimensional process $(X(t))$ defined on a probability space $(\Omega, \mathcal{F}, P)$ satisfies that: for any $s,t\in[0,1]$
$$E(\|X(t)-X(s)\|^{\alpha})\le c|t-s|^{1+\varepsilon}.$$
Then, there exists a continuous version $X(t)$ whose path are $\gamma$-holder for any $\gamma\in[0,\varepsilon/\alpha),$ i.e.,
$$\|X(t)-X(s)\|\le c_0 |t-s|^{\gamma}.$$
By Facts 1 and Fact 2, for any $p>0$
$$E(|B_H(t)-B_H(s)|^p)=|t-s|^{pH}E(|B_H(1)|^p):=c|t-s|^{1+(pH-1)}.$$
Then, by Kolmogorov’s continuity criterion, put $\gamma=(pH-1)/p$ and letting $p\to \infty$ we get the desired result.
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原文地址：http://www.cnblogs.com/jinjun/p/5256587.html
