Strong convergence rate of Runge--Kutta methods and simplified step-$N$ Euler schemes for SDEs driven by fractional Brownian motions
For researchers working on numerical methods for SDEs with fractional noise, this work provides rigorous convergence rates and resolves an open conjecture, though the contribution is incremental as it extends existing techniques.
This paper establishes the optimal strong convergence rate of Runge-Kutta methods and simplified step-N Euler schemes for SDEs driven by fractional Brownian motions with Hurst parameter H in (1/2,1), achieving a rate of 2H-1/2. It resolves a conjecture from the literature for this range of H.
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 $H\in(\frac12,1)$. 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 $2H-\frac12$. 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 $H\in(\frac12,1)$. Numerical experiments verify the theoretial convergence rate.