Fractional Brownian motion

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a>. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a <a href="/facts/Continuous-time_stochastic_process/HDwC9wg8">continuous-time</a> <a href="/facts/Gaussian_process/MrBq7kYW">Gaussian process</a> 
 
 
 
 
 B
 
 H
 
 
 (
 t
 )
 
 
 {\textstyle B_{H}(t)}
 
 on 
 
 
 
 [
 0
 ,
 T
 ]
 
 
 {\textstyle [0,T]}
 
, that starts at zero, has <a href="/facts/Expectation_(mathematics)/1XV0JKL8">expectation</a> zero for all 
 
 
 
 t
 
 
 {\displaystyle t}
 
 in 
 
 
 
 [
 0
 ,
 T
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 {\textstyle [0,T]}
 
, and has the following <a href="/facts/Covariance_function/pqUTwANG">covariance function</a>:

E
        [
        
          B
          
            H
          
        
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          B
          
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        =
        
          
            
              1
              2
            
          
        
        (
        
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    {\displaystyle E[B_{H}(t)B_{H}(s)]={\tfrac {1}{2}}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),}

where H is a real number in (0, 1), called the <a href="/facts/Hurst_exponent/kyxonWEh">Hurst index</a> or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968).
The value of H determines what kind of process the fBm is:

<ul><li>if H = 1/2 then the process is in fact a <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> or <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a>;</li>
<li>if H > 1/2 then the increments of the process are positively <a href="/facts/Correlation/egkluAEm">correlated</a>;</li>
<li>if H < 1/2 then the increments of the process are negatively correlated.</li></ul>
Fractional Brownian motion has <a href="/facts/Stationary_increments/m9A3YQW8">stationary increments</a> X(t) = BH(s+t) − BH(s) (the value is the same for any s). The increment process X(t) is known as fractional Gaussian noise.
There is also a generalization of fractional Brownian motion: n-th order fractional Brownian motion, abbreviated as n-fBm. n-fBm is a Gaussian, <a href="/facts/Self-similar/9Nvo7edY">self-similar</a>, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm is classical fBm.
Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist <a href="/facts/Robert_Brown_(botanist%2c_born_1773)/Yf68N7tb">Robert Brown</a>; fractional Gaussian noise is named after mathematician <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>.

Fractional Brownian motion open-in-new

Fractional Brownian motion