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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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    1. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    2. Stoev, Stilian A. & Taqqu, Murad S., 2006. "How rich is the class of multifractional Brownian motions?," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 200-221, February.
    3. Dieter Sondermann, 2006. "Introduction to Stochastic Calculus for Finance," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-34837-5, July.
    4. Samorodnitsky, Gennady, 1991. "Probability tails of Gaussian extrema," Stochastic Processes and their Applications, Elsevier, vol. 38(1), pages 55-84, June.
    5. Casse, Jérôme & Marckert, Jean-François, 2016. "Processes iterated ad libitum," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3353-3376.
    6. Azmoodeh, Ehsan & Sottinen, Tommi & Viitasaari, Lauri & Yazigi, Adil, 2014. "Necessary and sufficient conditions for Hölder continuity of Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 230-235.
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