Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

TL;DR

This paper investigates the existence and approximation of pathwise Stieltjes integrals involving discontinuous functions of stochastic processes, providing new conditions, estimates, and a change of variables formula.

Contribution

It introduces a notion of variability for processes to ensure integrability of discontinuous functionals and establishes convergence and approximation results for these integrals.

Findings

Pathwise integrals exist under new variability conditions.

Numerical approximation errors are quantitatively estimated.

A change of variables formula for these integrals is derived.

Abstract

In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and H\"older 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 $t \mapsto f(X_t)$ 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 $\int f(X_t) \, dX_t$.