论文标题
$ l^{q} $ - 近似随机向量函数近似的错误估计值
$L^{q}$-error estimates for approximation of irregular functionals of random vectors
论文作者
论文摘要
Avikainen表明，对于任何$ p，q \ in [1，\ infty）$，以及$ \ mathbb {r} $中的任何功能$ f $ funiation $ f $，它认为$ \ mathbb {e} [| f（x） \ Mathbb {在本文中，我们将为$ \ Mathbb {r}^{d} $，orlicz-sobolev space，sobolev Space，带有可变指数的sobolev空间和sobolev sobolev space的空间提供此估算的多维版本。我们论点的主要思想是使用这些功能空间的耐铁木最大估计和尖锐的特征。我们应用主要结果来分析随机微分方程解的某些不规则函数的数值近似值。
Avikainen 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.
