Backward Euler method for stochastic differential equations with non-Lipschitz coefficients

TL;DR

This paper analyzes the backward Euler method for stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, proving its order of convergence and validating optimality through numerical experiments.

Contribution

It establishes the order of convergence for the backward Euler method in this stochastic setting and confirms its optimality via asymptotic error analysis and numerical validation.

Findings

Backward Euler method has order 1 for these SDEs.

The convergence rate is proven to be optimal.

Numerical experiments support theoretical results.

Abstract

We study the traditional backward Euler method for $m$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H > 1/2$ whose drift coefficient satisfies the one-sided Lipschitz condition. The backward Euler scheme is proved to be of order $1$ and this rate is optimal by showing the asymptotic error distribution result. Two numerical experiments are performed to validate our claims about the optimality of the rate of convergence.