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 that is easier to implement.

本研究的目的是找到一种通用方法，用于生成给定随机微分方程解的路径，该方法比具有高斯增量的标准Euler-Maruyama方案更有效。首先，我们用时间间隔的一般划分来刻画欧拉-丸山方案中路径误差的渐近分布，然后证明误差减少了一个因子$（d+2）/d$当使用与球体的击打时间相关联的隔板来驱动d维布朗运动时。在对称的分区类别中，此减少比率是最好的。接下来，我们表明，通过使用易于实现的运动球体的击球时间，可以实现接近最佳的降低。