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Using Fractional Brownian Motion and ARFIMA for Hurst-Based Simulations

Article Quant Q&A · Author: Ice
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Summary

The document responds to a question about simulating processes with different Hurst exponent values for research on mean reversion. It identifies fractional Brownian motion as a process parameterized by the Hurst exponent and points to simulation code as a way to generate sample paths. It also suggests exploring ARFIMA processes, which can model long-range dependence.

The questioner reports that a cumulative random walk simulation produced a Hurst estimate near one half, consistent with a standard random-walk benchmark. The response offers candidate process families but does not explain how to estimate the Hurst exponent, select parameters, or interpret values above or below particular thresholds. Fractional Brownian motion and ARFIMA provide avenues for simulation, but the document does not establish that a given Hurst value alone identifies a profitable mean-reversion strategy.

Key ideas

  • Fractional Brownian motion is parameterized by the Hurst exponent and can generate simulated paths.
  • A cumulative random walk is associated in the document with a Hurst estimate near one half.
  • ARFIMA processes are suggested for investigating long-range dependence.
  • The response does not specify estimation procedures or prove that Hurst thresholds imply mean reversion.

Tags

StatisticsMean reversionMachine learning
Full text
# What are the equation that gives hurst exponent of value >0.7 and <0.3?


# What are the equation that gives hurst exponent of value >0.7 and <0.3?












I had been working on algorithm which uses the Hurst Exponent. Once i random walk simulation on matlab, x = cumsum(randm(1000,1)), I was able to get a hurst value close to 0.5.

To analyze the use of Hurst Exponent in mean aversion or mean reversion, may i know what are the equation models that i can used to run Monte Carlo simulations?

## Answer by vanguard2k (score 4)

https://quant.stackexchange.com/a/10615

The corresponding process would be fractional brownian motion (see here)

It is parametrized by the Hurst Exponent.

On the referenced site you find a link to some matlab code for simulating realizations of fractional BM.

If you want to see some fractional Gaussian Noise in action (Matlab) you can do so here.

Further more you might want to look into ARFIMA processes...

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.