represents fractional Brownian motion process with drift μ, volatility σ, and Hurst index h.


represents fractional Brownian motion process with drift 0, volatility 1, and Hurst index h.



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Basic Examples  (3)

Simulate a fractional Brownian motion process:

Mean and variance functions:

Covariance function:

Scope  (11)

Basic Uses  (6)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for different Hurst indices:

Process parameter estimation:

Correlation function:

Absolute correlation function:

Process Slice Properties  (5)

Univariate SliceDistribution:

First-order probability density function for the slice distribution:

Multivariate slice distributions:

Second-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis are constant:


Generating functions:

CentralMoment and its generating function:


Cumulant and its generating function:

Generalizations & Extensions  (1)

Useful shortcut evaluates to its full form counterpart:

Properties & Relations  (4)

FractionalBrownianMotionProcess is not weakly stationary:

Fractional Brownian motion does not have independent increments for :

Compare to the product of expectations:

Conditional cumulative probability distribution:

WienerProcess is a special case of fractional Brownian motion:

Compare mean functions:

Compare covariance functions:

Compare univariate slice distributions:

Neat Examples  (3)

Simulate a fractional Brownian motion process in two dimensions:

Compare 3D behavior of fractional Brownian motion depending on the Hurst parameter:

Simulate 500 paths from a fractional Brownian motion process:

Take a slice at 1 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1:

Introduced in 2012