We combine the rough path theory and stochastic backward error analysis to develop a new framework for error analysis on numerical schemes. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter $H\in(\frac14,\frac12)$ is $(2H-\frac12)^-$ for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the result of the corresponding implementable approximation of the L\'evy area of fractional Brownian motion. Our result gives a positive answer to the conjecture proposed in [11] for the case $H\in(\frac13,\frac12)$, and reveals for the first time that numerical schemes constructed by a second-order Taylor expansion do converge for the case $H\in(\frac14,\frac13]$.
翻译:我们把粗路径理论和后向错误分析结合起来,为数字方法的错误分析制定新的框架。根据我们的方法,我们证明,由多倍多倍多维分数运动驱动的修改的微沙分方程Milstein方案与Hurst参数$H\in(frac14,\frac12)的几乎肯定的汇合率是(2H-\frac12)$-美元,用于足够顺畅的系数,这是最佳的,因为它与分数布朗运动的L\'evy区域相应的可执行近似结果相一致。我们的结果对[11]中为案件提议的$H\in(frac13,\frac12)$的推测作出了积极的答复,并首次显示,用第二顺序泰勒扩展构建的数值方案对案件$H\in(frac14,\frac13)美元进行合并。