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Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions
IMA Journal of Numerical Analysis ( IF 2.1 ) Pub Date : 2020-07-24 , DOI: 10.1093/imanum/draa019
Jialin Hong 1 , Chuying Huang 1 , Xu Wang 1
Affiliation  

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.

中文翻译:

分数布朗运动驱动的随机微分方程两类方案的最优收敛速度

本文研究了由Hurst参数| $ H \ in(\ frac 12,1)$ |⁠的多维分数布朗运动(fBms)驱动的随机微分方程的数值方案。基于数值解对行驶噪声的连续依赖性,我们针对强收敛速度| $ 2H- \ frac 12 $ |⁠提出了Runge–Kutta方法的阶条件,这是逼近Lévy的最佳强收敛速度fBms的面积。通过提供“阶段值”,我们提供了一种替代方法来分析显式方案的收敛速度,从而将方案解释为Runge-Kutta方法。利用此技术,简化的步长|| $ N $ |的强收敛速度。 得到了欧拉方案,它给出了一个猜想的答案 Deya等。(2012)当| $ H \ in(\ frac 12,1)$ |⁠时。数值实验验证了理论收敛速度。
更新日期:2020-07-24
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