Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

Abstract

In this article we study the existence of pathwise Stieltjes integrals of the form ∫ f(Xt) dYt for nonrandom, possibly discontinuous, evaluation functions f and Hölder continuous random processes X and Y. We discuss a notion of sufficient variability for the process X which ensures that the paths of the composite process t ↦ f(Xt) are almost surely regular enough to be integrable. We show that the pathwise integral can be defined as a limit of Riemann–Stieltjes sums for a large class of discontinuous evaluation functions of locally finite variation, and provide new estimates on the accuracy of numerical approximations of such integrals, together with a change of variables formula for integrals of the form ∫ f(Xt) dXt .