Some properties on fBm

Posted 2020-06-19

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.