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Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

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  • Chen, Zhe
  • Leskelä, Lasse
  • Viitasaari, Lauri

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.

Suggested Citation

  • Chen, Zhe & Leskelä, Lasse & Viitasaari, Lauri, 2019. "Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2723-2757.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2723-2757
    DOI: 10.1016/j.spa.2018.08.002
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