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Transactions of the American Mathematical Society Series B

ISSN 2330-0000

   
 
 

 

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


Authors: Rama Cont and Nicolas Perkowski
Journal: Trans. Amer. Math. Soc. Ser. B 6 (2019), 161-186
MSC (2010): Primary 60H05
DOI: https://doi.org/10.1090/btran/34
Published electronically: April 10, 2019
MathSciNet review: 3937343
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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 ($C^{p-1}$) 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.


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Additional Information

Rama Cont
Affiliation: Mathematical Institute, University of Oxford, Oxford, United Kingdom; and LPSM, CNRS-Sorbonne Université
MR Author ID: 651275
Email: Rama.Cont@maths.ox.ac.uk

Nicolas Perkowski
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Leipzig & Humboldt–Universität zu Berlin
MR Author ID: 999469

Received by editor(s): April 18, 2018
Received by editor(s) in revised form: September 28, 2018
Published electronically: April 10, 2019
Additional Notes: N. Perkowski is grateful for the kind hospitality at University of Technology Sydney where this work was completed, and for financial support through the Bruti-Liberati Scholarship. N. Perkowski also gratefully acknowledges financial support by the DFG via the Heisenberg Program and Research Unit FOR 2402.
Article copyright: © Copyright 2019 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)