Optimal approximation rate of certain stochastic integrals

TL;DR

This paper characterizes the approximation rates of certain stochastic integrals based on the integrability of a function H, providing conditions for optimal convergence rates in terms of n.

Contribution

It offers a detailed analysis of the approximation rates of stochastic integrals, linking these rates to the integrability properties of the function H involved.

Findings

Characterization of when A_n(H) ≤ c/√n in terms of H's integrability.

Conditions for A_n(H) ≤ c/√n^β and ≥ 1/(c√n^β) based on H.

Application of results to approximate specific stochastic integrals.

Abstract

Given an increasing function $H:[0,1)\to [0,\infty)$ and $$ A_n(H):=\inf_{\tau\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} (t_i-t)H^2(t)dt)^{{1/2}}, $$ where $\mathcal{T}_n:=\{\tau=(t_i)_{i=0}^n: 0=t_0<t_1<...<t_n=1\}$, we characterize the property $A_n(H)\leq \frac{c}{\sqrt{n}}$, and give conditions for $A_n(H)\leq \frac{c}{\sqrt{n^\beta}}$ and $A_n(H)\geq \frac{1}{c\sqrt{n^\beta}}$ for $\beta\in (0,1)$, both in terms of integrability properties of $H$. These results are applied to the approximation of certain stochastic integrals.