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.
In Avikainen (2009, On irregular functionals of SDEs and the Euler scheme. Finance Stoch., 13, 381–401) the author showed that, for any $p,q \in [1,\infty )$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $ \mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}} $, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is an arbitrary random variable. In this article we will provide multi-dimensional versions of this estimate for functions of bounded variation in $\mathbb{R}^{d}$, Orlicz–Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy–Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.
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