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Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity. (English) Zbl 1411.60082

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.

MSC:

60H05 Stochastic integrals
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
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