# FractionalBrownianMotionProcess

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.

# Examples

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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:

Wolfram Research (2012), FractionalBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.

#### Text

Wolfram Research (2012), FractionalBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.

#### CMS

Wolfram Language. 2012. "FractionalBrownianMotionProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.

#### APA

Wolfram Language. (2012). FractionalBrownianMotionProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html

#### BibTeX

@misc{reference.wolfram_2024_fractionalbrownianmotionprocess, author="Wolfram Research", title="{FractionalBrownianMotionProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_fractionalbrownianmotionprocess, organization={Wolfram Research}, title={FractionalBrownianMotionProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html}, note=[Accessed: 12-September-2024 ]}