Fractional Ornstein–Uhlenbeck (fOU) — MATLAB/Octave

Tools to simulate a 1-D fractional Brownian motion (fBm), build an fOU process, and reproduce estimator diagnostics (almost sure convergence + asymptotic distribution). Code runs in MATLAB or GNU Octave.

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Contents

├─ Almostsureconvergence_test.m # smoke/diagnostic test (lightweight)

├─ Asymptoticdistribution_test.m # smoke/diagnostic test (lightweight)

├─ eta_quantities.m # main helper for estimator (was: testest)

├─ fbm1d.m # Davies–Harte fBm sampler (1D)

├─ Newton2.m # small Newton solver used by tests

└─ (other .m files as needed)

Requirements

• MATLAB R2020a+ (recommended), or
• GNU Octave ≥ 8.0 (CI uses 8.4)

No external toolboxes are required.

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Quick start

MATLAB

% In MATLAB, from the repo root
addpath(genpath(pwd));           % add all .m files to the path

% Run the light tests
run('Almostsureconvergence_test.m');
run('Asymptoticdistribution_test.m');

% Example: compute estimator ingredients at level s with stepsize h
[result] = eta_quantities(8, 0.01);   % returns [H2, H3]

Main functions

eta_quantities(s, h) -> [H2, H3]

Computes the summary quantities used by the estimator from a simulated fOU path.

• s — level (uses n = 2^s coarse steps)
• h — macro step size Internally simulates fBm via fbm1d, builds the fOU with parameters set at the top of the file (edit H, Theta, sigma there).

fbm1d(H, N, T) -> B

Generates a single-path 1D fBm of length N+1 on [0, T] with Hurst H using the Davies–Harte method.

• H ∈ (0,1), N time steps, T horizon Returns a column vector B with N+1 points and B(1)=0.

Newton2(...)

Minimal Newton solver used by the tests. See the function header for arguments.

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Performance tips

The tests are configured light for CI. To run heavier experiments locally:

• In eta_quantities.m, you can increase N (fine sub-steps), s (so n=2^s), or Monte-Carlo loops (if you add them) for higher fidelity.
• Long runs scale roughly linearly with N * n. Start small (s=7..9, moderate N) and grow as needed.

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Reproducibility

fbm1d.m uses randn internally. For reproducible runs, set the RNG before calls:

rng(42, 'twister');   % MATLAB
% or in Octave:
rand("seed", 42); randn("seed", 42);

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CI

The repository includes a GitHub Actions workflow that runs the light tests on Octave to ensure the code compiles and the basic pipeline works. Heavier loops are commented out to keep CI fast.

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License

MIT. See LICENSE.

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Citation

If you use this code in academic work, please cite the repository:

@misc{FractionalOrnstein,
  author = {El Mehdi Haress, Yaozhong Hu},
  title  = {Fractional Ornstein–Uhlenbeck (fOU) MATLAB/Octave code},
  year   = {2025},
  url    = {https://github.com/ElMehdiHaress/FractionalOrnstein}
}

Authors

Yaozhong Hu and El Mehdi Haress