Mathematics > Probability
[Submitted on 16 Jan 2020 (v1), last revised 6 Mar 2020 (this version, v2)]
Title:A generalized Avikainen's estimate and its applications
View PDFAbstract: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 Hölder 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.
Submission history
From: Dai Taguchi [view email][v1] Thu, 16 Jan 2020 01:40:32 UTC (45 KB)
[v2] Fri, 6 Mar 2020 05:01:58 UTC (48 KB)
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