Approximations of fractional Brownian motion

TL;DR

This paper introduces a novel approximation method for fractional Brownian motion using a two-parameter Poisson process, expanding existing approaches that typically use one-parameter processes.

Contribution

It presents a new approximation technique for fractional Brownian motion employing a two-parameter Poisson process, with rigorous proof of convergence.

Findings

Successful construction of the approximation method.

Proof of tightness and finite-dimensional distribution convergence.

Potential for improved modeling of fractional Brownian motion.

Abstract

Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the one-parameter fractional Brownian motion is constructed using a two-parameter Poisson process. The proof involves the tightness and identification of finite-dimensional distributions.