A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion

TL;DR

This paper proposes a new second-order numerical scheme for stochastic differential equations driven by multidimensional fractional Brownian motion with Hurst parameter > 1/3, avoiding Levy area calculations.

Contribution

It introduces an implementable Milstein-type scheme that replaces Levy area terms with products of increments, validated through rough paths theory and error analysis.

Findings

Scheme achieves convergence without Levy area terms

Utilizes rough paths techniques for analysis

Provides practical implementation for fBm-driven SDEs

Abstract

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.