An efficient weak Euler-Maruyama type approximation scheme of very high dimensional SDEs by orthogonal random variables

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 scheme proposed in this paper, in contrast, can deal with this problem by generating only single uniform random number at every time step. The scheme saves the time for simulation of very high dimensional SDEs. In particular, we will show that Euler-Maruyama approximation generated by the Walsh system is efficient in high dimensions.