TWO QUADRATURE RULES FOR STOCHASTIC IT(O)over-cap-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,W-P(0,T),sigma is an element of(0,2), P is an element of[2,infinity). We introduce two quadrature rules: The first is best suited for the parameter range sigma is an element of(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 is an element of(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.