Rate of convergence for discretization of integrals with respect to Fractional Brownian motion

TL;DR

This paper establishes the $L^r$ convergence and provides a rate of convergence for the uniform discretization of stochastic integrals with respect to fractional Brownian motion, extending previous almost sure convergence results.

Contribution

It introduces the $L^r$ convergence analysis and derives a convergence rate for discretizing integrals with fractional Brownian motion, for a broad class of convex functions.

Findings

Proves $L^r$ convergence of discretized stochastic integrals.

Derives a convergence rate approaching $H - 1/2$.

Extends previous almost sure convergence results.

Abstract

In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$ is considered. In Statistics & Decisions, 27, 129-143, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as possible to $H - 1/2$.