Avikainen provided a sharp upper bound of the difference $\mathbb{E}[|g(X)-g(\widehat{X})|^{q}]$ by the moments of $|X-\widehat{X}|$ for any one-dimensional random variables $X$ with bounded density and $\widehat{X}$, and function of bounded variation $g$. In this article, we generalize this estimate to any one-dimensional random variable $X$ with H\"older continuous distribution function. As applications, we provide the rate of convergence for numerical schemes for solutions of one-dimensional stochastic differential equations (SDEs) driven by Brownian motion and symmetric $\alpha$-stable with $\alpha \in (1,2)$, fractional Brownian motion with drift and Hurst parameter $H \in (0,1/2)$, and stochastic heat equations (SHEs) with Dirichlet boundary conditions driven by space--time white noise, with irregular coefficients. We also consider a numerical scheme for maximum and integral type functionals of SDEs driven by Brownian motion with irregular coefficients and payoffs which are related to multilevel Monte Carlo method.

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广义 Avikainen 估计及其应用

Avikainen 通过 $|X-\widehat{X} 的矩提供了差值 $\mathbb{E}[|g(X)-g(\widehat{X})|^{q}]$ 的尖锐上限|$ 用于任何具有有界密度和 $\widehat{X}$ 的一维随机变量 $X$，以及有界变化函数 $g$。在本文中，我们将这个估计推广到任何具有 H\"旧连续分布函数的一维随机变量 $X$。作为应用，我们提供了一维随机微分方程 (SDE) 解的数值方案的收敛速度) 由布朗运动和对称 $\alpha$-stable with $\alpha \in (1,2)$、分数布朗运动与漂移和 Hurst 参数 $H \in (0,1/2)$ 和随机热驱动具有由时空白噪声驱动的狄利克雷边界条件的方程 (SHE)，具有不规则系数。