This paper deals with asymptotic errors, limit theorems for errors between numerical and exact solutions of stochastic differential equation (SDE) driven by one-dimensional fractional Brownian motion (fBm). The Euler-Maruyama, higher-order Milstein, and Crank-Nicolson schemes are among the most studied numerical schemes for SDE (fSDE) driven by fBm. Most previous studies of asymptotic errors have derived specific asymptotic errors for these schemes as main theorems or their corollary. Even in the one-dimensional case, the asymptotic error was not determined for the Milstein or the Crank-Nicolson method when the Hurst exponent is less than or equal to $1/3$ with a drift term. We obtained a new evaluation method for convergence and asymptotic errors. This evaluation method improves the conditions under which we can prove convergence of the numerical scheme and obtain the asymptotic error under the same conditions. We have completely determined the asymptotic error of the Milstein method for arbitrary orders. In addition, we have newly determined the asymptotic error of the Crank-Nicolson method for $1/4<H\leq 1/3$.

翻译：本文研究由一维分数布朗运动驱动的随机微分方程数值解与精确解之间误差的渐近性质及极限定理。Euler-Maruyama格式、高阶Milstein格式和Crank-Nicolson格式是分数布朗运动驱动随机微分方程中最受关注的数值格式。以往大多数关于渐近误差的研究都将这些格式的特定渐近误差作为主要定理或其推论给出。即使在一维情形下，当Hurst指数小于等于$1/3$且存在漂移项时，Milstein方法和Crank-Nicolson方法的渐近误差仍未确定。我们提出了一种新的收敛性与渐近误差评估方法。该方法改进了证明数值格式收敛性所需的条件，并能在相同条件下获得渐近误差。我们完整确定了任意阶Milstein方法的渐近误差。此外，我们首次确定了Crank-Nicolson方法在$1/4<H\leq 1/3$区间内的渐近误差。