Sampling fractional Brownian motion in presence of absorption: a Markov Chain method

TL;DR

This paper introduces a Monte Carlo method to efficiently generate fractional Brownian motion with absorption at the origin, validating analytical predictions about endpoint distributions for various Hurst exponents.

Contribution

It develops a Markov Chain sampling technique for fBm with absorption, enabling large-scale simulations and analysis of endpoint distributions for small H values.

Findings

Distribution of endpoints follows a power law with exponent (1-H)/H.

Method accurately reproduces analytical finite-length corrections for H=0.5.

Numerical results align with theoretical predictions for various H.

Abstract

We study fractional Brownian motion (fBm) characterized by the Hurst exponent H. Using a Monte Carlo sampling technique, we are able to numerically generate fBm processes with an absorbing boundary at the origin at discrete times for a large number of 10^7 time steps even for small values like H=1/4. The results are compatible with previous analytical results that the distribution of (rescaled) endpoints y follow a power law P(y) y^\phi with \phi=(1-H)/H, even for small values of H. Furthermore, for the case H=0.5 we also study analytically the finite-length corrections to the first order, namely a plateau of P(y) for y->0 which decreases with increasing process length. These corrections are compatible with the numerical results.