Self-similarity

The process is self-similar, since in terms of probability distributions:

${\displaystyle B_{H}(at)\sim |a|^{H}B_{H}(t).}$

This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a fractal property. FBm can also be defined as the unique mean-zero Gaussian process, null at the origin, with stationary and self-similar increments.

Stationary increments

It has stationary increments:

${\displaystyle B_{H}(t)-B_{H}(s)\;\sim \;B_{H}(t-s).}$

Long-range dependence

For H > ½ the process exhibits long-range dependence,

${\displaystyle \sum _{n=1}^{\infty }E[B_{H}(1)(B_{H}(n+1)-B_{H}(n))]=\infty .}$

Regularity

Sample-paths are almost nowhere differentiable. However, almost-all trajectories are locally Hölder continuous of any order strictly less than H: for each such trajectory, for every T > 0 and for every ε > 0 there exists a (random) constant c such that

${\displaystyle |B_{H}(t)-B_{H}(s)|\leq c|t-s|^{H-\varepsilon }}$

for 0 < s,t < T.

Dimension

With probability 1, the graph of B_H(t) has both Hausdorff dimension^[2] and box dimension of 2−H.

Integration

As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not semimartingales.

Frequency-domain interpretation

Just as Brownian motion can be viewed as white noise filtered by ${\displaystyle \omega ^{-1}}$ (i.e. integrated), fractional Brownian motion is white noise filtered by ${\displaystyle \omega ^{-H-1/2}}$ (corresponding to fractional integration).

Method 1 of simulation

One can simulate sample-paths of an fBm using methods for generating stationary Gaussian processes with known covariance function. The simplest method relies on the Cholesky decomposition method of the covariance matrix (explained below), which on a grid of size ${\displaystyle n}$ has complexity of order ${\displaystyle O(n^{3})}$. A more complex, but computationally faster method is the circulant embedding method of Dietrich & Newsam (1997).

Suppose we want to simulate the values of the fBM at times ${\displaystyle t_{1},\ldots ,t_{n}}$ using the Cholesky decomposition method.

• Form the matrix ${\displaystyle \Gamma ={\bigl (}R(t_{i},\,t_{j}),i,j=1,\ldots ,\,n{\bigr )}}$ where ${\displaystyle \,R(t,s)=(s^{2H}+t^{2H}-|t-s|^{2H})/2}$.
• Compute ${\displaystyle \,\Sigma }$ the square root matrix of ${\displaystyle \,\Gamma }$, i.e. ${\displaystyle \,\Sigma ^{2}=\Gamma }$. Loosely speaking, ${\displaystyle \,\Sigma }$ is the "standard deviation" matrix associated to the variance-covariance matrix ${\displaystyle \,\Gamma }$.
• Construct a vector ${\displaystyle \,v}$ of n numbers drawn independently according to a standard Gaussian distribution,
• If we define ${\displaystyle \,u=\Sigma v}$ then ${\displaystyle \,u}$ yields a sample path of an fBm.

In order to compute ${\displaystyle \,\Sigma }$, we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of ${\displaystyle \,\Gamma }$:

• Since ${\displaystyle \,\Gamma }$ is symmetric, positive-definite matrix, it follows that all eigenvalues ${\displaystyle \,\lambda _{i}}$ of ${\displaystyle \,\Gamma }$ satisfy ${\displaystyle \,\lambda _{i}>0}$, (${\displaystyle i=1,\dots ,n}$).
• Let ${\displaystyle \,\Lambda }$ be the diagonal matrix of the eigenvalues, i.e. ${\displaystyle \Lambda _{ij}=\lambda _{i}\,\delta _{ij}}$ where ${\displaystyle \delta _{ij}}$ is the Kronecker delta. We define ${\displaystyle \Lambda ^{1/2}}$ as the diagonal matrix with entries ${\displaystyle \lambda _{i}^{1/2}}$, i.e. ${\displaystyle \Lambda _{ij}^{1/2}=\lambda _{i}^{1/2}\,\delta _{ij}}$.

Note that the result is real-valued because ${\displaystyle \lambda _{i}>0}$.

• Let ${\displaystyle \,v_{i}}$ an eigenvector associated to the eigenvalue ${\displaystyle \,\lambda _{i}}$. Define ${\displaystyle \,P}$ as the matrix whose ${\displaystyle i}$-th column is the eigenvector ${\displaystyle \,v_{i}}$.

Note that since the eigenvectors are linearly independent, the matrix ${\displaystyle \,P}$ is invertible.

• It follows then that ${\displaystyle \Sigma =P\,\Lambda ^{1/2}\,P^{-1}}$ because ${\displaystyle \Gamma =P\,\Lambda \,P^{-1}}$.

Method 2 of simulation

It is also known that ^[4]

${\displaystyle B_{H}(t)=\int _{0}^{t}K_{H}(t,s)\,dB(s)}$

where B is a standard Brownian motion and

${\displaystyle K_{H}(t,s)={\frac {(t-s)^{H-{\frac {1}{2}}}}{\Gamma (H+{\frac {1}{2}})}}\;_{2}F_{1}\left(H-{\frac {1}{2}};\,{\frac {1}{2}}-H;\;H+{\frac {1}{2}};\,1-{\frac {t}{s}}\right).}$

Where ${\displaystyle _{2}F_{1}}$ is the Euler hypergeometric integral.

Say we want to simulate an fBm at points ${\displaystyle 0=t_{0}<t_{1}<\cdots <t_{n}=T}$.

• Construct a vector of n numbers drawn according to a standard Gaussian distribution.
• Multiply it component-wise by √T/n to obtain the increments of a Brownian motion on [0, T]. Denote this vector by ${\displaystyle (\delta B_{1},\ldots ,\delta B_{n})}$.
• For each ${\displaystyle t_{j}}$, compute

${\displaystyle B_{H}(t_{j})={\frac {n}{T}}\sum _{i=0}^{j-1}\int _{t_{i}}^{t_{i+1}}K_{H}(t_{j},\,s)\,ds\ \delta B_{i}.}$

The integral may be efficiently computed by Gaussian quadrature.