Fractional Brownian Motion

Discrete-time Recipe for Fractional Brownian Motion

Intuition

Fractional Brownian motion (fBM) is a generalization of Brownian motion with parameter \( H \in (0,1) \) (the Hurst parameter) that controls the roughness and memory of the process. Its covariance kernel is $$ K(s, t) = \frac{1}{2} \left( s^{2H} + t^{2H} - |s-t|^{2H} \right) $$ For \( H = 1/2 \), this reduces to standard Brownian motion. For \( H > 1/2 \), increments are positively correlated (persistent), and for \( H < 1/2 \), increments are negatively correlated (anti-persistent).

General Recipe

1. Choose the Hurst parameter \( H \in (0,1) \).
2. Generate the desired covariance matrix \( \Gamma_{ij} = K(t_i, t_j) \) for a grid of time points \( t_i \).
3. Decompose \( \Gamma = Q \Lambda Q^T \) (eigendecomposition or Cholesky).
4. Generate a Gaussian vector \( Z \sim N(0, I_n) \).
5. Set \( X = Q \Lambda^{1/2} Z \) (or use Cholesky: \( X = L Z \)).

Python Implementation

import numpy as np

def fbm_discrete(steps=100, H=0.5, paths=1, T=1):
    t = np.linspace(0, T, steps)
    Gamma = np.zeros((steps, steps))
    for i in range(steps):
        for j in range(steps):
            Gamma[i, j] = 0.5 * (t[i]**(2*H) + t[j]**(2*H) - abs(t[i]-t[j])**(2*H))
    L = np.linalg.cholesky(Gamma + 1e-10*np.eye(steps))
    Z = np.random.normal(0, 1, (steps, paths))
    X = L @ Z
    return X.T, t

Summary: This approach produces fBM sample paths with the correct covariance structure by constructing the covariance matrix and using Cholesky decomposition.