Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions

This paper investigates numerical schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions (fBms) with Hurst parameter $H\in (\frac 12,1)$. Based on the continuous dependence of numerical solutions on the driving noises, we propose the order conditions of Runge–Kutta methods for the strong convergence rate $2H-\frac 12$, which is the optimal strong convergence rate for approximating the Lévy area of fBms. We provide an alternative way to analyse the convergence rate of explicit schemes by adding ‘stage values’ such that the schemes are interpreted as Runge–Kutta methods. Taking advantage of this technique the strong convergence rate of simplified step-$N$ Euler schemes is obtained, which gives an answer to a conjecture in Deya et al. (2012) when $H\in (\frac 12,1)$. Numerical experiments verify the theoretical convergence rate.