We will introduce Euler–Maruyama approximations given by an orthogonal system in $L^2[0,1]$ for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs to generate uniform random numbers at every time step in numerical simulation. The schemes proposed in this paper, in contrast, can deal with this problem by generating very few uniform random numbers at every time step. The schemes save time in the simulation of very high dimensional SDEs. In particular, we conclude that an Euler–Maruyama approximation based on the Walsh system is efficient in high dimensions.

我们将介绍由正交系统给出的Euler–Maruyama近似 ${\mathrm{大号}}^{\mathrm{2个}}[0，\mathrm{1个}]$对于高维SDE，可能是SPDE的有限维近似。通常，维数越大，在数值模拟的每个时间步上生成统一随机数的需求就越大。相比之下，本文提出的方案可以通过在每个时间步生成很少的统一随机数来解决此问题。该方案节省了模拟高维SDE的时间。特别是，我们得出的结论是，基于Walsh系统的Euler-Maruyama逼近在高维方面是有效的。