  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.
