Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity

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.

Keywords

stochastic integration, quadrature rules, fractional Sobolev spaces, Sobolev–Slobodeckij norm

2010 Mathematics Subject Classification

60H05, 60H35, 65C30

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The authors wish to express their gratitude to Stefan Heinrich for many interesting discussions on this topic. This research was carried out in the framework of Matheon supported by Einstein Foundation Berlin. The second named author also gratefully acknowledges financial support by the German Research Foundation through the research unit FOR 2402 – Rough paths, stochastic partial differential equations and related topics – at TU Berlin.

Received 17 January 2018

Accepted 13 August 2018

Published 18 April 2019