2020/05/07 by Dai Taguchi, Taguchi, Dai, Akihiro Tanaka +3
Economics, Econometrics and Finance · Mathematics · #26A45 #60H35 #65C05 #65C30 #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2005.03219
openalex publication_date 2020/05/07 · openalex created_date 2020/12/07 · openalex updated_date 2026/07/28
Avikainen showed that, for any p,q ∈ [1,∞), and any function f of bounded variation in ℝ, it holds that 𝔼[|f(X)-f(\widehatX)|q] ≤ C(p,q) 𝔼[|X-\widehatX|p](1)/(p+1), where X is a one-dimensional random variable with a bounded density, and \widehatX is an arbitrary random variable. In this article, we will provide multi-dimensional versions of this estimate for functions of bounded variation in ℝd, 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.