# Fractional Ornstein-Uhlenbeck Process

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The first two moments (mean and variance) of an Ornstein–Uhlenbeck (OU) process are approximated with stochastic expansions (linear combinations of iterated integrals of the paths). The first three parameters are the usual parameters for an OU process: a high mean reversion makes the convergence to the mean faster and a high volatility increases the variance. The Hurst index controls the roughness of the fractional Brownian motion: the higher the value, the smoother the path. At Hurst index 0.5, this reduces to the usual Brownian motion.

[more]
Contributed by: Christophe Ladroue (October 2010)

Based on a program by: Christophe Ladroue

After work by: Anastasia Papavasiliou and Christophe Ladroue

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

References

[1] T. J. Lyons, M. J. Caruna, and T. Lévy, "Differential Equations Driven by Rough Paths*,*" in *Ecole d'et*é* de Probabilit*é*s de Saint-Flour,* XXXIV, (J. Picard, ed.), Berlin: Springer, 2007.

[2] A. Papavasiliou and C. Ladroue, "Parameter Estimation for Rough Differential Equations*.*" (2010) http://arxiv.org/abs/0812.3102.

## Permanent Citation

"Fractional Ornstein-Uhlenbeck Process"

http://demonstrations.wolfram.com/FractionalOrnsteinUhlenbeckProcess/

Wolfram Demonstrations Project

Published: October 4 2010