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On Approximate Representation of Fractional Brownian Motion

Konstantin A. Rybakov

TL;DR

This work develops a Legendre-polynomial–based orthogonal expansion for fractional Brownian motion (FBM) to enable continuous-time approximation and simulation. By deriving explicit expansion coefficients $K_{ij}^H$ for the kernel $k_H$ in terms of Legendre-coefficient matrices and fractional-integral operators, it yields a practical pathwise representation $B_H(t)=\sum_i \mathcal B_i^H \hat P(i,t)$ with $\mathcal B^H=K^H\mathcal V$. The main result provides closed-form expressions for $K_{ij}^H$ applicable across $H<1/2$, $H>1/2$, and $H=1/2$, highlighting self-similarity through $T^{H+1/2}$ scaling. The paper also introduces finite-term approximations with exact mean-square error formulas, enabling controllable accuracy via truncation level $L$ and study of convergence. These contributions offer a stable, exact, continuous-time alternative to existing FBM simulation methods and can extend to FBM-driven processes and related functionals.

Abstract

This paper considers the orthogonal expansion of the fractional Brownian motion relative to the Legendre polynomials. Such an expansion has not only theoretical but also practical interest, since it can be applied to approximate and simulate the fractional Brownian motion in continuous time. The relations for the mean square approximation error are presented, and a comparison with the previously obtained result is carried out.

On Approximate Representation of Fractional Brownian Motion

TL;DR

This work develops a Legendre-polynomial–based orthogonal expansion for fractional Brownian motion (FBM) to enable continuous-time approximation and simulation. By deriving explicit expansion coefficients for the kernel in terms of Legendre-coefficient matrices and fractional-integral operators, it yields a practical pathwise representation with . The main result provides closed-form expressions for applicable across , , and , highlighting self-similarity through scaling. The paper also introduces finite-term approximations with exact mean-square error formulas, enabling controllable accuracy via truncation level and study of convergence. These contributions offer a stable, exact, continuous-time alternative to existing FBM simulation methods and can extend to FBM-driven processes and related functionals.

Abstract

This paper considers the orthogonal expansion of the fractional Brownian motion relative to the Legendre polynomials. Such an expansion has not only theoretical but also practical interest, since it can be applied to approximate and simulate the fractional Brownian motion in continuous time. The relations for the mean square approximation error are presented, and a comparison with the previously obtained result is carried out.

Paper Structure

This paper contains 6 sections, 2 theorems, 70 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The Problem Statement
  3. Preliminary Results
  4. The Main Result
  5. Approximate Representation of Fractional Brownian Motion
  6. Conclusions

Key Result

Theorem 1

Let $K^H$ be the infinite matrix with elements $K_{ij}^H$ defined by Equation eqSpKH, $H \in (0,1)$. Then, for the $j$th column of the matrix $K^H$, we have where $(3/2-H)^{\overline{k}}$ is the upper factorial, and elements of the infinite column matrix $F^{H+1/2+k}$ are given by Equations eqSpFAlphaExplicit--eqSpFAlphaImplicitAlpha. For $H \neq 1/2$, values $K_{ij}^H$ satisfy the following equa

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3