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

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Rama Cont and Nicolas Perkowski
**HTML**| PDF - Trans. Amer. Math. Soc. Ser. B
**6**(2019), 161-186

## 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.
- © Copyright 2019 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B
**6**(2019), 161-186 - MSC (2010): Primary 60H05
- DOI: https://doi.org/10.1090/btran/34
- MathSciNet review: 3937343