Fractional Brownian motion

A normalized fractional Brownian motion (denoted fBm) $B^H(t)$ on $[0,T]\; ,\; Tin\; mathbb\{R\}$ is a continuous-time Gaussian process starting at zero, with mean zero, and having the following correlation function:

:$E\; [B^H(t)\; B^H(s)]\; =frac\{1\}\{2\}\; (|t|^\{2H\}+|s|^\{2H\}-|t-s|^\{2H\}),$

where $H$, called the Hurst index or Hurst parameter associated to the fractional Brownian motion, is a real number in $[0,1]$.

The value of $H$ determines what kind of process the fBm is:
* if $H=1/2$, the process is in fact a regular Brownian motion;
* if $H>1/2$, the increments of the process are positively correlated;
* if , the increments of the process are negatively correlated.

Properties

elf-similarity

The process is said to be self-similar, since in terms of distributions:::$B^H(at)sim\; |a|^\{H\}B^H(t).$

Stationary increments

It has stationary increments:::$B^H(t)\; -\; B^H(s)sim\; B^H(t-s).$

Long-range dependence

For $H>1/2$, the process exhibits long-range dependence, which means that ::$sum\_\{n=1\}^infty\{E\; [B^H(1)(B^H(n+1)-B^H(n))]\; \}=infty.$

Regularity

Sample-paths are almost nowhere differentiable. Precisely, almost-all trajectories are Hölder continuous of any order strictly less than $H$: for each trajectory, there exists a constant $c$ such that::$|B^H(t)-B^H(s)|\; le\; c\; |t-s|^\{H-varepsilon\}$for any $varepsilon\; >0$.

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 Martingales.

ample paths

Practical computer realisations of fBm can be generated, although obviously they cannot really be fractal. The sample paths chosen can be thought of as showing discrete sampled points on an fBm process. Three realisations are shown below, each with 1000 points of fBm with Hurst parameter 0.75.

Two realisations are shown below, each showing 1000 points of fBm, the first with Hurst parameter 0.95 and the second with Hurst parameter 0.55.

Method 1 of simulation :

One can simulate sample-paths of fBm as any Gaussian process of known covariance. Say we aim to have simulated values at $t\_1,\; cdots,\; t\_n$.

* Form the matrix where $R(t,s)=(s^\{2H\}+t^\{2H\}-|t-s|^\{2H\})/2$.

* Compute a square root of $Gamma$, say $sigma$. Use for instance the Cholesky decomposition method.

* Construct a vector of $n$ numbers drawn according a standard gaussian distribution.

* Apply $sigma$ to this vector yields a sample-path of fBm.

Method 2 of simulation :

It is also known that $B^H(t)=int\_0^t\; K\_H(t,s)dB(s)$ where $B$ is a standard Brownian motion and $K\_H(t,s)=frac\{1\}\{Gamma(H+1/2)\}(t-s)^\{H-1/2\};\_2F\_1(H-1/2;,\; 1/2-H;;\; H+1/2;,\; 1-t/s).$See section on Euler Hypergeometric integral for the definition of $\_2F\_1$.

Say we want simulate fBm at points .

* Construct a vector of $n$ numbers drawn according a standard gaussian distribution.

* Multiply it component-wise by sqrt(T/n) to obtain the increments of a Brownian motion on [0,T] . Denote this vector by $(delta\; B\_1,\; cdots,\; delta\; B\_n)$.

* For each $t\_j$, compute$B^H(t\_j)=frac\{n\}\{T\}sum\_\{i=0\}^\{j-1\}\; int\_\{t\_i\}^\{t\_\{i+1\; K\_H(t\_j,,\; s),\; ds\; delta\; B\_i.$

The integral may be efficiently computed by Gaussian quadrature. Hypergeometric functions are part of the GNU scientific library .

See also

* Multifractal: The generalized framework of fBm.

References

:*Beran, J. (1994) "Statistics for Long-Memory Processes", Chapman & Hall. ISBN 0-412-04901-5.