Strong convergence rate for two classes of implementable methods for SDEs driven by fractional Brownian motions

Abstract. We investigate 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 ∈ ( 2 , 1). These two classes of numerical schemes are implementable in the sense that the required information from the driving noises are only their increments. We prove the solvability of implicit Runge–Kutta methods and the continuous dependence of their continuous versions with respect to the driving noises in Hölder semi-norm. Based on these results, the order conditions are proposed for the strong convergence rate 2H − 1 2 , which is caused by the approximation of the Lévy area type processes. We get the same strong convergence rate for simplified step-N Euler schemes by comparing them with an explicit Runge–Kutta method. This gives an answer to the conjecture in [5] for H ∈ ( 2 , 1).