Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Chen, Zhe¹  Leskela, Lasse²  Viitasaari, Lauri³ 

[1] NYU, Tandon Sch Engn, Dept Finance & Risk Engn, New York, NY 10003 USA

[2] Aalto Univ, Sch Sci, Dept Math & Syst Anal, Espoo, Finland

[3] Aalto Univ, Sch Elect Engn, Dept Commun & Networking, Espoo, Finland

关键词: Composite stochastic process;    Generalised Stieltjes integral;    Fractional calculus;    Riemann-Liouville integral;    Fractional Sobolev space;    Gagliardo-Slobodeckij seminorm;    Fractional Sobolev-Slobodeckij space;    Bounded p-variation;   

DOI  :  10.1016/j.spa.2018.08.002

来源: Elsevier

【 摘 要 】

In this article we study the existence of pathwise Stieltjes integrals of the form integral f(X-t) dY(t) for nonrandom, possibly discontinuous, evaluation functions f and Holder 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 bar right arrow 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 integral f(X-t) dX(t). (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free