$L^{q}$-error estimates for approximation of irregular functionals of random vectors

TL;DR

This paper extends Avikainen's $L^{q}$-error estimates from one-dimensional to multi-dimensional settings, covering various function spaces, and applies these results to improve numerical approximations of irregular functionals in stochastic differential equations.

Contribution

It provides multi-dimensional $L^{q}$-error estimates for functions of bounded variation and other complex function spaces, using Hardy--Littlewood maximal estimates.

Findings

Extended error estimates to multi-dimensional function spaces.

Applied estimates to numerical approximation of irregular functionals.

Demonstrated effectiveness in stochastic differential equations.

Abstract

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.