  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.
