Euler scheme for SDEs driven by fractional Brownian motions: integrability and convergence in law

TL;DR

This paper analyzes the Euler scheme for stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than 1/3, establishing integrability and convergence in law with explicit rates.

Contribution

It proves the integrability of the Euler scheme and its Malliavin derivatives uniformly in step size, and derives the convergence rate in law for the scheme.

Findings

Euler scheme and derivatives are integrable uniformly in step size

Convergence rate in law is n^{1-4H+ε}

Uses Malliavin calculus and generalized greedy sequence argument

Abstract

In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size $n$. Then we use the integrability results to derive the convergence rate in law $n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums.