Spectral Representation and Simulation of Fractional Brownian Motion
Konstantin A. Rybakov
TL;DR
The paper addresses the challenge of simulating fractional Brownian motion $B_H(\cdot)$ in continuous time and introduces a new spectral representation based on Legendre polynomials. It derives explicit spectral characteristics for the kernel and for key operators, enabling the expression $\mathcal{B}^H=K^H\mathcal{V}$ and the covariance form $S^H=K^H[K^H]^T$, which together provide a principled framework for simulation. It then develops computationally stable algorithms to evaluate these spectral characteristics and constructs a finite, controllable approximation $\tilde{B}_H$ with explicit error components $\varepsilon_1$ and $\varepsilon_2$, supported by numerical experiments showing that Legendre bases yield superior accuracy over alternatives. The method offers a rigorous, scalable tool for continuous-time fBm simulation and provides a general framework applicable to fractional integration and stochastic differential equations in broader contexts.
Abstract
The paper gives a new representation for the fractional Brownian motion that can be applied to simulate this self-similar random process in continuous time. Such a representation is based on the spectral form of mathematical description and the spectral method. The Legendre polynomials are used as the orthonormal basis. The paper contains all the necessary algorithms and their theoretical foundation, as well as the results of numerical experiments.