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.
- FractionalBrownianMotionProcess is also known as fractal Brownian motion or fractional Wiener process.
- FractionalBrownianMotionProcess is a continuous-time and continuous-state random process.
- FractionalBrownianMotionProcess is a Gaussian process with mean function and covariance function . It reduces to a WienerProcess for .
- FractionalBrownianMotionProcess allows μ to be any real number, σ to be any positive real number, and h to be a real number between 0 and 1.
- FractionalBrownianMotionProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examplesopen allclose all
Basic Examples (3)
Basic Uses (6)
Process Slice Properties (5)
First-order probability density function for the slice distribution:
Multivariate slice distributions:
Compute the expectation of an expression:
Calculate the probability of an event:
Skewness and kurtosis are constant:
CentralMoment and its generating function:
Cumulant and its generating function:
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:
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:
Wolfram Research (2012), FractionalBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.
Wolfram Language. 2012. "FractionalBrownianMotionProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.
Wolfram Language. (2012). FractionalBrownianMotionProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html