Discretizing the fractional Levy area

TL;DR

This paper analyzes the discretization errors of the Levy area for fractional Brownian motion, revealing different convergence regimes depending on the Hurst parameter and deriving the asymptotic error distributions.

Contribution

It provides sharp bounds and exact convergence rates for Euler and trapezoidal schemes, including the asymptotic error distribution, for fractional Brownian motion Levy area.

Findings

Convergence rate for H<3/4 is n^{-2H+1/2}.

For H=3/4, convergence rate is n^{-1}(log(n))^{1/2}.

For H>3/4, convergence rate is n^{-1}.

Abstract

In this article, we give sharp bounds for the Euler- and trapezoidal discretization of the Levy area associated to a d-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean-square convergence rate of the Euler scheme. For H<3/4 the exact convergence rate is n^{-2H+1/2}, where n denotes the number of the discretization subintervals, while for H=3/4 it is n^{-1} (log(n))^{1/2} and for H>3/4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exact convergence rate n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic error distribution of the Euler scheme. For H lesser than 3/4 one obtains a Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type.