Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity

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Title: Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity
Authors: Cont, R
Perkowski, N
Item Type: Journal Article
Abstract: We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of p-th variation along a sequence of time partitions. For paths with finite p-th variation along a sequence of time partitions, we derive a change of variable formula for p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. We show that the pathwise integral satisfies an `isometry' formula in terms of p-th order variation and obtain a `signal plus noise' decomposition for regular functionals of paths with strictly increasing p-th variation. For less regular (Cp−1) functions we obtain a Tanaka-type change of variable formula using an appropriately defined notion of local time. These results extend to multidimensional paths and yield a natural higher-order extension of the concept of `reduced rough path'. We show that, while our integral coincides with a rough-path integral for a certain rough path, its construction is canonical and does not involve the specification of any rough-path superstructure.
Issue Date: 10-Apr-2019
Date of Acceptance: 7-Dec-2018
URI: http://hdl.handle.net/10044/1/65012
DOI: https://dx.doi.org/10.1090/btran/34
ISSN: 0002-9947
Publisher: American Mathematical Society
Start Page: 161
End Page: 186
Journal / Book Title: Transactions of the American Mathematical Society
Volume: 6
Copyright Statement: ©2019 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0) (https://creativecommons.org/licenses/by/3.0/)
Keywords: math.PR
math.PR
math.CA
math.PR
math.PR
math.CA
Publication Status: Published
Appears in Collections:Financial Mathematics
Mathematics
Faculty of Natural Sciences



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