We can define the error distribution as the limiting distribution of the error between the solution $Y$ of a given stochastic differential equation (SDE) and its numerical approximation $\hat{Y}^{(m)}$, weighted by the convergence rate between the two. A goal when studying the error distribution is to provide a way of determination for error distributions for any SDE and numerical scheme that converge to the exact solution. By dividing the error into a main term and a remainder term in a particular way, the author shows that the remainder term can be negligible compared to the main term under certain suitable conditions. Under these conditions, deriving the error distribution reduces to deriving the limiting distribution of the main term. Even if the dimension is one, there are unsolved problems about the asymptotic behavior of the error when the SDE has a drift term and $0<H\leq 1/3$, but our result in the one-dimensional case can be adapted to any Hurst exponent. The main idea of the proof is to define a stochastic process $Y^{m, \rho}$ with the parameter $\rho$ interpolating between $Y$ and $\hat{Y}^{(m)}$ and to estimate the asymptotic expansion for it. Using this estimate, we determine the error distribution of the ($k$)-Milstein scheme and of the Crank-Nicholson scheme in unsolved cases.

翻译：我们可以将误差分布定义为给定随机微分方程（SDE）的解$Y$与其数值逼近$\hat{Y}^{(m)}$之间误差的极限分布，并以两者之间的收敛速度进行加权。研究误差分布的一个目标是为任意收敛于精确解的SDE和数值格式提供误差分布的确定方法。通过以特定方式将误差分解为主项和余项，作者证明在适当条件下余项相较于主项可忽略。在这些条件下，推导误差分布可简化为推导主项的极限分布。即使在一维情形下，当SDE包含漂移项且$0<H\leq 1/3$时，误差的渐近行为仍存在未解决的问题，但我们在单变量情形下的结果可适用于任意赫斯特指数。证明的核心思想是定义具有插值参数$\rho$的随机过程$Y^{m, \rho}$，该过程在$Y$与$\hat{Y}^{(m)}$之间平滑过渡，并估计其渐近展开式。利用该估计，我们确定了（$k$）阶米尔斯坦格式和克兰克-尼科尔森格式在未解情形下的误差分布。