Optimal approximation of Skorohod integrals - examples with substandard rates

TL;DR

This paper investigates the optimal mean square error approximation rates for Itô and Skorohod integrals using equidistant Brownian motion discretization, revealing substandard rates under certain conditions.

Contribution

It provides new insights into the approximation rates of Skorohod integrals, including conditions for achieving optimal rates and examples of lower rates, extending existing knowledge beyond standard results.

Findings

Optimal approximation rates smaller than n^{-1} for certain integrals.

Discontinuities and Weyl equidistribution influence approximation rates.

Conditions for achieving n^{-1/2} rate in Skorohod integrals.

Abstract

We consider optimal approximation with respect to the mean square error of It\^o integrals and Skorohod integrals given an equidistant discretization of the Brownian motion. We obtain for suitable integrands optimal rates smaller than the standard $n^{-1}$, where $n$ denotes the number of evaluations of the Brownian motion. For the It\^o integral this is due to the Weyl equidistribution theorem and discontinuities of the integrand. For the Skorohod integral the situation is more complicated and relies on a reformulation of the Wiener chaos expansion. Here, we specify conditions on the integrands to obtain optimal rates $n^{-1/2}$, respectively, examples of lower rates.