The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion

TL;DR

This paper analyzes the convergence rates of Euler approximations for solutions to stochastic differential equations driven by fractional Brownian motion, providing explicit error bounds depending on the Hurst parameter.

Contribution

It establishes precise convergence rate estimates for Euler schemes applied to SDEs driven by fractional Brownian motion with Hurst index greater than 1/2.

Findings

Euler approximation converges at rate O(δ^{2H-1}) for pathwise SDEs

Discrete Skorohod-type equations converge at rate O(δ^H)

Convergence rates depend explicitly on the Hurst index H

Abstract

The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(\delta^{2H-1})$ ($\delta$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(\delta^H)$.