Two quadrature rules for stochastic It\^o-integrals with fractional Sobolev regularity

TL;DR

This paper develops two new quadrature rules for stochastic Itô integrals with fractional Sobolev regularity, achieving convergence order equal to the regularity parameter, and demonstrates their effectiveness through theoretical analysis and numerical experiments.

Contribution

It introduces two novel quadrature methods tailored for different fractional Sobolev regularity ranges, extending stochastic integral approximation techniques.

Findings

Both quadrature rules achieve convergence order equal to the Sobolev regularity parameter.

The Riemann-Maruyama approximation is effective for $\sigma ext{ in } (0,1)$.

Numerical experiments confirm the theoretical convergence rates.

Abstract

In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in $W^{\sigma,p}(0,T)$, $\sigma \in (0,2)$, $p \in [2,\infty)$. We introduce two quadrature rules: The first is best suited for the parameter range $\sigma \in (0,1)$ and consists of a Riemann-Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with $\sigma \in (1,2)$. In both cases the order of convergence is equal to $\sigma$ with respect to the $L^p$-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments.