2022年1月
$L^{q}$-error estimates for approximation of irregular functionals of random vectors
IMA Journal of Numerical Analysis
- ,
- ,
- 巻
- 42
- 号
- 1
- 開始ページ
- 840
- 終了ページ
- 873
- 記述言語
- 掲載種別
- 研究論文(学術雑誌)
- DOI
- 10.1093/imanum/draa096
- 出版者・発行元
- Oxford University Press (OUP)
<title>Abstract</title>
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.
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.
- リンク情報
- ID情報
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- DOI : 10.1093/imanum/draa096
- ISSN : 0272-4979
- eISSN : 1464-3642