Strong convergence rate of Runge--Kutta methods and simplified step-$N$ Euler schemes for SDEs driven by fractional Brownian motions

This paper focuses on the strong convergence rate of both Runge–Kutta methods and simplified step-$N$ Euler schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions with $H\in (\frac{1}{2},1)$. Based on the continuous dependence of both stage values and numerical schemes on driving noises, order conditions of Runge–Kutta methods are proposed for the optimal strong convergence rate $2H-\frac{1}{2}$. This provides an alternative way to analyze the convergence rate of explicit schemes by adding `stage values’ such that the schemes are comparable with Runge–Kutta methods. Taking advantage of this technique, the optimal strong convergence rate of simplified step-N Euler scheme is obtained, which gives an answer to a conjecture in $[3]$ when $H\in (\frac{1}{2},1)$. Numerical experiments verify the theoretial convergence rate.