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

Cont, Rama; Perkowski, Nicolas

doi:10.1090/btran/34

Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity
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by Rama Cont and Nicolas Perkowski;

Trans. Amer. Math. Soc. Ser. B 6 (2019), 161-186

DOI: https://doi.org/10.1090/btran/34

Published electronically: April 10, 2019

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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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