Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity
Provides theoretical convergence guarantees for numerical integration of stochastic processes with fractional regularity, benefiting computational mathematics and stochastic simulation.
This paper introduces two quadrature rules for stochastic Itô-integrals with integrands having fractional Sobolev regularity, achieving convergence order σ in L^p-norm. The methods are applied to integrate Poisson processes with discontinuous paths.
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 $W^{σ,p}(0,T)$, $σ\in (0,2)$, $p \in [2,\infty)$. We introduce two quadrature rules: The first is best suited for the parameter range $σ\in (0,1)$ 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 $σ\in (1,2)$. In both cases the order of convergence is equal to $σ$ 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.