## Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

*Stochastic Processes and their Applications*, to appear.- doi:10.1016/j.spa.2018.08.002
- arXiv:1612.00498

### Abstract

In this article we study the existence of pathwise Stieltjes integrals of the form *∫ f(X _{t}) dY_{t}* 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(X*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

_{t})*∫ f(X*.

_{t}) dX_{t}