Optimal Approximation of Stochastic integrals with Respect to a Homogeneous Poisson Process

Abstract

We consider strong approximation in the asymptotic setting of stochastic integrals with respect to a homogeneous Poisson process. An integrand \({f:[0,T] \to \mathbb{R}}\) is assumed to be an r-smooth function from \({\mathcal{C}^r([0,T])}\). We show that the \({L^p}\)-error (\({p \in [1,+\infty)}\)) of any approximation method, which uses n values of f, cannot converge to zero faster than \({n^{-r}}\) as \({n\to+\infty}\). This holds except for a subset of \({\mathcal{C}^r([0,T])}\) with empty interior. The best speed of convergence is achieved by the Ito–Taylor algorithm.