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

TL;DR

This paper establishes the optimal strong convergence rates for Runge--Kutta and simplified step-N Euler schemes applied to SDEs driven by fractional Brownian motions with Hurst parameter H in (1/2, 1), providing theoretical insights and numerical validation.

Contribution

It introduces order conditions for Runge--Kutta methods to achieve the optimal convergence rate and confirms the conjectured rate for simplified step-N Euler schemes in this context.

Findings

Runge--Kutta methods achieve convergence rate 2H-1/2.

Simplified step-N Euler schemes also attain the optimal rate.

Numerical experiments confirm theoretical convergence rates.

Abstract

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.