A generalized Avikainen's estimate and its applications

TL;DR

This paper extends Avikainen's estimate to a broader class of random variables with H"older continuous distribution functions and applies it to analyze convergence rates of numerical schemes for various stochastic differential equations and related functionals.

Contribution

It generalizes Avikainen's estimate to H"older continuous distribution functions and demonstrates its applications in convergence analysis of numerical schemes for complex stochastic systems.

Findings

Established convergence rates for SDE solutions driven by Brownian motion and stable processes.

Analyzed stochastic heat equations with irregular coefficients.

Provided error bounds for numerical schemes of SDE functionals.

Abstract

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.