Eisenmann, Monika; Kruse, Raphael Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity. (English) Zbl 1411.60082 Commun. Math. Sci. 16, No. 8, 2125-2146 (2018). Summary: 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. Cited in 3 Documents MSC: 60H05 Stochastic integrals 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations Keywords:stochastic integration; quadrature rules; fractional Sobolev spaces; Sobolev-Slobodeckij norm PDFBibTeX XMLCite \textit{M. Eisenmann} and \textit{R. Kruse}, Commun. Math. Sci. 16, No. 8, 2125--2146 (2018; Zbl 1411.60082) Full Text: DOI arXiv