Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

Zhe Chen, Lasse Leskelä and Lauri Viitasaari

Stochastic Processes and their Applications, 2019, vol. 129, issue 8, 2723-2757

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.

Keywords: Composite stochastic process; Generalised Stieltjes integral; Fractional calculus; Riemann–Liouville integral; Fractional Sobolev space; Gagliardo–Slobodeckij seminorm; Fractional Sobolev–Slobodeckij space; Bounded p-variation (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:129:y:2019:i:8:p:2723-2757

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DOI: 10.1016/j.spa.2018.08.002

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