Efficient discretisation of stochastic differential equations

The aim of this study is to find a generic method for generating a path of
the solution of a given stochastic differential equation which is more
efficient than the standard Euler-Maruyama scheme with Gaussian increments.
First we characterize the asymptotic distribution of pathwise error in the
Euler-Maruyama scheme with a general partition of time interval and then, show
that the error is reduced by a factor (d+2)/d when using a partition associated
with the hitting times of sphere for the driving d-dimensional Brownian motion.
This reduction ratio is the best possible in a symmetric class of partitions.
Next we show that a reduction which is close to the best possible is achieved
by using the hitting time of a moving sphere which is easier to implement.